1 Introduction

The present paper adopts the definition of Rhodius [12] for the spectrum \(\sigma _{\mathbb {F}}(T)\) and continues to define the numerical range \(W_{\mathbb {F}}(T)\) for any bounded linear operator T acting on a Hilbert space H over the real or complex field \({\mathbb {F}}\); it then proves that, in case \(T=T^*\), \(\sigma _{\mathbb {R}}(T)\cap \{-||T||,||T||\}\ne \emptyset\) and \(\mathrm{co}(\sigma _{\mathbb {R}}(T))=\overline{W_{\mathbb {R}}(T)}\subset [-||T||,||T||]\). The results are well-known in case \({\mathbb {F}}={\mathbb {C}}\) and the independent proofs of the general case given here provide novel short proofs for the known results. This obliges us to acknowledge that the paper is partly expository. The classical properties of complex selfadjoint operators are proven here with arguments not depending on the theory of complex functions such as Liouville’s theorem in the proof of \(\sigma (T)\ne \emptyset\). However, the results are not valid for real normal operators N as it is clear from the \(2\times 2\) real normal operator \(N:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\) defined by

$$\begin{aligned} N[1,0]^T=[0,1]^T~\mathrm{and~}~N[0,1]^T=[-1,0]^T \end{aligned}$$
(1.1)

that \(\sigma _{\mathbb {R}}(N)=\emptyset \ne W_{\mathbb {R}}(N)=\{0\}\).

In Sect. 2, certain topological as well as geometrical properties of \(\sigma _{\mathbb {F}}(T)\) and \(W_{\mathbb {F}}(T)\) are proven. Let \((H,{\mathbb {F}})\) be as in Sect. 2 and assume \(N=T+i\tau\) for some selfadjoint operators \(T,\tau \in B(H)\), where \(\tau\) is necessarily 0 if \({\mathbb {F}}={\mathbb {R}}\). The results of Sect. 2 are applied to the pair \((N,{\mathbb {F}})\) to provide a new proof of the known Borel functional calculus for N in which the sophisticated Gelfand–Naimark theorem [9] is avoided and the Berberian’s amalgamation theory [2] is shortened via a double integration method. (See (3.7)–(3.10) below.) The latter results, in turn, are used to sharpen the bounded Borel functional calculus \(f\mapsto f(N):{\mathcal {A}}_b(N)\rightarrow B(H)\) to an isometric Borel functional calculus \(f\mapsto f(N):L^\infty _{\mathbb {F}}(N)\rightarrow B(H)\), where \({\mathcal {A}}_b(N)\subset {\mathbb {F}}^{\mathbb {F}}\) denotes the algebra of all Borel functions bounded on \(\sigma _{\mathbb {F}}(N)\) and \(L^\infty _{\mathbb {F}}(N)\) is the completion of \({\mathcal {A}}_b(N)\) with respect to a norm to be defined in Sect. 3. Our improvement of the Borel functional calculus also includes an essential extension of the known spectral mapping theorem \(f(\sigma _{\mathbb {F}}(N))=\sigma _{\mathbb {F}}(f(N))\) for continuous functions to the one for the classes \({\mathcal {A}}_b(N)\) and \(L^\infty _{\mathbb {F}}(N)\).

With each type of the field (\({\mathbb {R}}\) or \({\mathbb {C}}\)), we may be dealing with separable or nonseparable Hilbert spaces. When the symbol \({\mathbb {F}}\) is used, we are dealing with a unified proof in which we ignore the type of the field as well as the cardinality of the orthonormal basis of the space. As a general rule, any result involving \(\sigma _{\mathbb {F}}(T)\) or \(W_{\mathbb {F}}(T)\) is a new result if \({\mathbb {F}}={\mathbb {R}}\). Theorem 3.1 is known (see [9,10,11]) but the proof contains a novel proof of the Borel functional calculus as described above. No matter what the field \({\mathbb {F}}\) be, Theorem 3.2 as well as its proof are completely new. We mention that the polynomial spectral mapping theorem \(f(\sigma _{\mathbb {C}}(N))= \sigma _{\mathbb {C}}(f(N))\) may fail to be true if \({\mathbb {C}}\) is replaced by \({\mathbb {R}}\); this is revealed by examining the simple \(2\times 2\) normal matrix N given at the beginning of the paper; however, we prove it for any real selfadjoint operator. (See Remark 1.2 and Theorem 2.1 below.)

In Sect. 4, we study the spectral measures of real selfadjoint operators. As usual, the arguments remain valid for complex normal operators. Here, the spectral measures are applied to the above-mentioned algebras \(L^\infty _{\mathbb {F}}(N)\). For more details on Borel measures, Borel functional calculus, and spectral measures, we may refer to [3,4,5,6].

Section 5 deals with (real or complex) normal operators whose domains are generated by their eigenvalues. We first have a different look at direct sums of infinitely many mutually perpendicular subspaces and study extensions of Bessel’s inequality and Parseval equalities. We give a very simple definition of the direct sum of an arbitrary family of mutually perpendicular subspaces and then use it to extend the notion of functional calculus for discrete normal operators. A similar extension is obtained for the spectral measure of such operators.

Let us fix some conventions which will be used throughout the paper. In \({\mathbb {R}}\), we feel free to use expressions such as (a) the polar decomposition \(\lambda =e^{i\theta }|\lambda |\), (b) the complex conjugate \(\lambda ^*=(\alpha +i\beta )^*=\alpha -i\beta\), (c) the real part \({\mathfrak {R}e}(\lambda )\), (d) the imaginary part \({\mathfrak {I}m}(\lambda )\), as well as the potentially real expressions (e) \(i^2\), (f) 0i, (g) i0, and (h) \(e^{ik\pi }\), provided that they be interpreted as (a’) \(\lambda =\pm |\lambda |\), (b’) \(\lambda = \lambda ^*=\alpha \pm i0=\alpha\), (c’) \({\mathfrak {R}e}(\lambda )=\lambda\), (d’) \({\mathfrak {I}m}(\lambda )=0\), (e’) \(i^2=-1\), (f’) \(0i=0\), (g’) \(i0=0\) and (h’) \(e^{ik\pi }=(-1)^k\) for \(k\in {\mathbb {Z}}\), respectively. We will keep the notation \({\bar{\gamma }}\) to denote the closure of a set \(\gamma\); in conformity with the notation \(a^*\) in a general \(C^*\)-algebra, the complex conjugate of a number \(\lambda \in {\mathbb {F}}\) will be denoted by \(\lambda ^*\). If f is any \({\mathbb {F}}\)-valued function, \(f^*\) denotes its complex conjugate; that is, \(f^*(t)=(f(t))^*={\mathfrak {R}e}(f(t))-i{\mathfrak {I}m}(f(t))\). Also, if \(\varDelta\) is a subset of a topological linear space, then \({\bar{\vee }}(\varDelta )\) will denote the closure of its linear span. (We define \({\bar{\vee }}(\emptyset )=\{0\}\).)

For an algebra \({\mathcal {A}}\) of \({\mathbb {F}}\)-valued functions, we set \({\mathcal {A}}^*=\{f^*:~f\in {\mathcal {A}}\}\). Then, \({\mathcal {A}}\) is called a \(*\)-algebra if \({\mathcal {A}}={\mathcal {A}}^*\). In particular, if \({\mathbb {F}}={\mathbb {R}}\), then \({\mathcal {A}}={\mathcal {A}}^*\) for every \({\mathcal {A}}\). The notation \({\mathfrak {B}}(={\mathfrak {B}}_L)\) stands for the \(\sigma\)-algebra of all Borel subsets of \(\sigma _{\mathbb {F}}(L)\). Also, the notation \({\mathcal {D}}(L)\), \({\mathcal {K}}(L)\), and \({\mathcal {R}}(L)\) are fixed to denote the domain, the kernel, and the range of an operator L.

The present introductory section will be concluded with certain properties of the Rhodius spectrum \(\sigma _{\mathbb {F}}(L)\) of a general \(L\in B({\mathcal {X}})\), where \({\mathcal {X}}\) is a Banach space over the real or complex field \({\mathbb {F}}\). The same will be done for the numerical range \(W_{\mathbb {F}}(L)\) when L is a Hilbert space operator. Immediate properties of these sets follow the definitions. The case \({\mathcal {X}}=\{0\}\) causes ambiguity; even if we assume without loss of generality that \({\mathcal {X}}\ne \{0\}\), the zero subspaces will surprise us in the middle of the arguments. So, we may take care of the ambiguities as follows.

Definition 1.1

For \(L\in B({\mathcal {X}})\), define its spectrum \(\sigma _{\mathbb {F}}(L)\) as follows.

  • Case 1. \({\mathcal {X}}\ne \{0\}\).

    $$\begin{aligned} \sigma _{\mathbb {F}}(L):={\mathbb {F}}\backslash \{\lambda \in {\mathbb {F}}:~\lambda I-L~\mathrm{~is~bijective}\}. \end{aligned}$$
    (1.2)

  • Case 2. \({\mathcal {X}}=\{0\}\). We have \(B({\mathcal {X}})=\{0\}\) and define

    $$\begin{aligned} \sigma _{\mathbb {F}}(L)=\emptyset . \end{aligned}$$
    (1.3)

Remark 1.1

To justify the case \({\mathcal {X}}=\{0\}\), note that the operator \(L=0\) is the only linear operator on \({\mathcal {X}}\), \(~0^{-1}=0=I\in B({\mathcal {X}})~\) and, in this particular case, \(\sigma _{\mathbb {F}}(0)=\emptyset\). Note that if \(\mathcal {\mathcal {X}}\ne \{0\}\), then \(0^{-1}\) does not exist and \(\sigma _{\mathbb {F}}(0)=\{0\}\).

For a scholarly investigation of the definition and properties of the Rhodius spectrum and numerical range of an operator [12] used and verified in the following formulas (1.4)–(1.9), we may refer to Theorem 1.1 of Appell et al. [1]; however, although not explicitly mentioned, we believe the authors of [1] assume the underlying field is complex; otherwise, the statement and the proof of Theorem 1.1 of [1] would contain errors. The theorem claims the validity of the spectral mapping theorem in the real case for which we have a counterexample. (See Remark 1.2 below.) Another error is the application of Liouville’s theorem which has no context in real analysis. Therefore, not being able to find a suitable reference, we present a proof which works for both real and complex fields. Of course, instead of the full strength of the spectral mapping theorem \(f(\sigma _{\mathbb {F}}(L))=\sigma _{\mathbb {F}}(f(L))\) claimed in Theorem 1.1 of [1], we only prove the weaker formula \(f(\sigma _{\mathbb {F}}(L))\subset \sigma _{\mathbb {F}}(f(L))\). (See 1.5). )

Theorem 1.1

The spectrum \(\sigma _{\mathbb {F}}(L)\) defined by (1.2)–(1.3) satisfies

$$\begin{aligned} \overline{\sigma _{\mathbb {F}}(L)}=\sigma _{\mathbb {F}}(L)\subset \{\lambda \in {\mathbb {F}}:~|\lambda |\le ||L||\}, \end{aligned}$$
(1.4)

and

$$\begin{aligned} f(\sigma _{\mathbb {F}}(L))\subset \sigma _{\mathbb {F}}(f(L))~{\mathrm{~ for~ all~}} f\in {\mathbb {F}}[x]. \end{aligned}$$
(1.5)

The latter inclusion can be sharpened to equality, if \({\mathbb {F}}={\mathbb {C}}\).

Proof

The boundedness of \(\sigma _{\mathbb {F}}(L)\) follows from the fact that \(L_\lambda (\lambda I-L)x=(\lambda I-L)L_\lambda x\equiv x\), where

$$\begin{aligned} L_\lambda x:= & {} \sum _{n=0}^\infty \lambda ^{-n-1}L^nx~, \end{aligned}$$
(1.6)
$$\begin{aligned} ||L_\lambda x||\le & {} \frac{1}{|\lambda |-||L||}||x||,~\forall x\in {\mathcal {X}}, ~\forall \lambda \in {\mathbb {F}}~\mathrm{~satisfying~}~ |\lambda |>||L||; \end{aligned}$$
(1.7)

here, the power series is absolutely convergent. This proves that \(\sigma _{\mathbb {F}}(L)\subset \{\lambda \in {\mathbb {F}}:~|\lambda |\le ||L||\}\). The closedness of \(\sigma _{\mathbb {F}}(L)\) follows from the fact that if \(\lambda _0\in {\mathbb {F}}\backslash \sigma _{\mathbb {F}}(L)\), then to every \(\lambda\) in an (open) neighborhood of radius r centered at \(\lambda _0\) there corresponds an operator \(R_\lambda\) satisfying \(R_\lambda (\lambda I-L)x=(\lambda I-L)R_\lambda x\equiv x\) and \(||R_\lambda x||\le \rho _\lambda ||x||\) for all \(x\in {\mathcal {X}}\), where \(R_\lambda\), \(\rho _\lambda\) and r are defined as follows:

$$\begin{aligned} R_\lambda x= & {} -\sum _{n=0}^\infty (\lambda -\lambda _0)^n(L-\lambda _0I)^{-n-1}x, \end{aligned}$$
(1.8)
$$\begin{aligned} \rho _\lambda= & {} \frac{1}{r-|\lambda -\lambda _0|},~\mathrm{~and~} ~r=||(L-\lambda _0I)^{-1}||^{-1} \end{aligned}$$
(1.9)

Again, here, the power series is absolutely convergent. In particular, \(~\sigma _{\mathbb {F}}(L)\) is a compact subset of the interval/disk \(\{\lambda \in {\mathbb {F}}:~|\lambda |\le ||L||\}\) in \({\mathbb {F}}\). This proves (1.4). To prove (1.5), let \(\mu \in f(\sigma _{\mathbb {F}}(L))\). Then \(\mu =f(\lambda )\) for some \(\lambda \in \sigma _{\mathbb {F}}(L)\). Hence, \(\mu -f(t)=(\lambda -t)g(t)\) for some \(g\in {\mathbb {F}}[x]\). Since \(\lambda I-L\) is not invertible, it follows that \(\mu I-f(L)=(\lambda I-L)g(L)\) is not invertible and that \(\mu \in \sigma _{\mathbb {F}}(f(L))\); i.e., \(f(\sigma _{\mathbb {F}}(L))\subset \sigma _{\mathbb {F}}(f(L))\). To show the desired converse, let \(\mu \in \sigma _{\mathbb {C}}(f(L))\) and factorize \(\mu I-f(L)\) as

$$\begin{aligned} (\mu I-f(L))=(\lambda _1I-L)(\lambda _2I-L)\ldots (\lambda _kI-L) ~\mathrm{~for~some~}~\lambda _1,\lambda _2,\ldots ,\lambda _k\in {\mathbb {C}}; \end{aligned}$$

note that at least one of the factors should be singular, say \(\lambda _i\in \sigma _{\mathbb {C}}(L)\). Thus \(\mu =f(\lambda _i)\in f(\sigma _{\mathbb {C}}(L))\).\(\square\)

Remark 1.2

Let \(N:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\) be as in (1.1). \([\sigma _{\mathbb {R}}(N)]^2=\emptyset \ne \{-1\}=\sigma _{\mathbb {R}}(N^2)\). Thus, the inclusion in (1.5) cannot be sharpened, and the equality 1.14 in Theorem 1.1 of [1] is not true in the real case.

Corollary 1.1

For a general operator \(L\in B({\mathcal {X}})\) and a point \(\lambda _0\notin \sigma _{\mathbb {F}}(L)\),

$$\begin{aligned}&||L||\ge \max |\sigma _{\mathbb {F}}(L)|:=\max \{|\lambda |:~\lambda \in \sigma _{\mathbb {F}}(L)\}~\mathrm{~and~}\\&||(L-\lambda _0I)^{-1}|| \ge 1/ \mathrm{dist}(\lambda _0,\sigma _{\mathbb {F}}(L)). \end{aligned}$$

Corollary 1.2

If U is a unitary operator on a Hilbert space H, then \(\sigma (U)\subset \{z\in {\mathbb {F}}:~|z|=1\}\).

Another concept needed in the spectral theory is the notion of the numerical range \(W_{\mathbb {F}}(L)\) for any Hilbert space operator \(L\in B(H)\):

$$\begin{aligned} W_{\mathbb {F}}(L):=\{\langle Lx,x\rangle :~x\in H,~||x||=1\}. \end{aligned}$$
(1.10)

The following theorem is well known in the complex case; the proof here is readjusted to cover the real case, too. We found no author to be particularly interested in the real case, though, we believe there must be some proof somewhere. Hence, for reader’s convenience, we present the following short unified proof which works for both \({\mathbb {R}}\) and \({\mathbb {C}}\).

Theorem 1.2

The numerical range of a Hilbert space operator \(L\in B(H)\) is a convex subset of \(~{\mathbb {F}}\) when the latter is identified as one of the Euclidean spaces \({\mathbb {R}}\) or \({\mathbb {R}}^2\).

Proof

Let \(u,v\in H\) be any two linearly independent unit vectors such that \(\langle Lu,u\rangle =\lambda \ne \langle Lv,v\rangle =\mu\). We must show that the open line segment joining \(\lambda\) and \(\mu\) lies in \(W_{\mathbb {F}}(L)\). Replace L by \((\mu -\lambda )^{-1}(L-\lambda I)\) to assume without loss of generality that \(\lambda =0\) and \(\mu =1\). Let \(L=A+B\) be the decomposition of L into its (obviously) selfadjoint part \(A=(L+L^*)/2\) and its skew selfadjoint part \(B=L-A\); the latter in the sense that \(\langle Bx,y\rangle =-\langle x,By\rangle\). Let \(x=zv+u\) for \(z\in {\mathbb {F}}\). Then, \(||x||>0\), \(~\langle Bu,v\rangle =-\langle u,Bv\rangle =-\overline{\langle Bv,u\rangle }\), and

$$\begin{aligned} \langle Lx,x\rangle= & {} \langle L(zv+u),zv+u\rangle \\= & {} |z|^2+z\langle Av,u\rangle +{\bar{z}} \langle Au,v\rangle +z \langle Bv,u\rangle + {\bar{z}}\langle Bu,v\rangle \\= & {} |z|^2+2~{\mathfrak {R}e}(z\langle Av,u\rangle )+2i~{\mathfrak {I}m}(z\langle Bv,u\rangle ). \end{aligned}$$

Consider the polar decomposition \(\langle Bv,u\rangle )=e^{i\theta }|\langle Bv,u\rangle )|\). (If \({\mathbb {F}}={\mathbb {R}}\), then \(\theta =0\) or \(\pi\).) Let \(z=te^{-i\theta }\) \((t\in {\mathbb {R}})\), let \(b=2~{\mathfrak {R}e}(e^{-i\theta }\langle Av,u\rangle )\), let \(b'=2~{\mathfrak {R}e}(e^{-i\theta } \langle v,u\rangle )\) and let

$$\begin{aligned} \phi (t)=\frac{t^2+bt}{t^2+b't+1}=\frac{\langle Lx,x\rangle }{||x||^2},~\forall ~t\in {\mathbb {R}}. \end{aligned}$$

Then, \(\phi (t)\) is a continuous rational function traveling in \(W_{\mathbb {F}}(L)\) and covering the line segment (0, 1) as t runs in \((0,\infty )\).\(\square\)

2 Polynomial spectral mapping theorem

In this section, we prove the spectral mapping theorem for real or complex selfadjoint operators and, as a consequence, we show that the spectrum of any selfadjoint operator is nonempty. Of course, this is well known in the complex case and is proven here for the first time for real case. We first state a lemma whose proof in the complex case is well known and works for the real case, too.

Lemma 2.1

Let \(N\in B(H)\) be a real or complex normal operator. Then \(\sigma _{\mathbb {F}}(N)\) consists of approximate point spectra. Moreover, \(\sigma _{\mathbb {F}}(N)\subset \overline{W_{\mathbb {F}}(N)}\). In particular, if \(T=T^*\), then \(\sigma _{\mathbb {F}}(T)\subset \overline{W_{\mathbb {F}}(T)}\subset {\mathbb {R}}\).

Theorem 2.1

If \(T\in B(H)\) is a real or complex selfadjoint operator, then

$$\begin{aligned} \sigma _{\mathbb {F}}(f(T))=f(\sigma _{\mathbb {F}}(T))\subset f({\mathbb {R}}),~ \forall ~ f\in {\mathbb {F}}[x]. \end{aligned}$$

Proof

Let \(f_m\in {\mathbb {F}}[x]\) be a prime monic polynomial of order m and define \(S_m=f_m(T)\). Then, \(f_m\) is necessarily of the form

$$\begin{aligned} f_m(t)=(t-\alpha )^m-(i\beta )^m,~t\in {\mathbb {F}}, ~\alpha ,\beta \in {\mathbb {R}};~m=1,2. \end{aligned}$$
(2.1)

Note that if \({\mathbb {F}}={\mathbb {C}}\), then \(m=1\); also, if \({\mathbb {F}}={\mathbb {R}}\), then either \(m=2\) in which case \(-(i\beta )^2=\beta ^2>0\) or \(m=1\) in which case \(\beta =0\).

In general, if \(u\in H\) is any unit vector, then

$$\begin{aligned} ||S_1u||^2= & {} ||S^*_1u||^2= \langle (T-\alpha I-i\beta I)u, (T-\alpha I-i\beta I)u\rangle \\= & {} \langle (T-\alpha I+i\beta I)(T-\alpha I-i\beta I)u,u\rangle = \langle [(T-\alpha I)^2+\beta ^2 I]u,u\rangle \\= & {} ||(T-\alpha I)u||^2+\beta ^2\ge \beta ^2, \end{aligned}$$

and

$$\begin{aligned} ||S_2u||= & {} ||S^*_2u||= \langle [ (T-\alpha I)^2+\beta ^2)]u,u\rangle = \langle (T-\alpha I)^2u,u\rangle +\langle \beta ^2u,u\rangle \\= & {} ||(T-\alpha I)u||^2+\beta ^2\ge \beta ^2. \end{aligned}$$

It follows that 0 is not in the approximate point spectrum of \(S_m~{\rm if~}\beta \ne 0\). Therefore, \(S_m\) is invertible if \(\beta \ne 0\). Now, consider the prime factorization \(\mu -f=g_1g_2\ldots g_k ~{\rm for~some~}\mu\in\mathbb F~{\rm and~ some}~f\in\mathbb F[x]\), where each \(g_i\) is a prime polynomial of the form \(f_m\) defined in (2.1). If \(\mu I-f(T)\) is singular, at least one of the factors \(g_i(T)\) is singular. We assume without loss of generality that \(i=k=1, g_1(t)=f_m(t)\) and \(f_m(T)\) is singular. Then, \(\beta = 0\) and, hence, \(m\ne 2\). It follows that \(g_1(T)=T-\alpha I\) and, thus, \(\alpha \in \sigma _{\mathbb {F}}(T)\). Therefore, \(\mu -f(\alpha )=0\) or, equivalently, \(\mu =f(\alpha )\in f(\sigma _{\mathbb {F}}(T))\). This, together with (1.5), completes the proof of the theorem.\(\square\)

The next theorem shows that, regardless of the type of the underlying field \({\mathbb {F}}\), the spectrum of a selfadjoint operator on a nonzero Hilbert space is nonempty. Here, the extreme points of the numerical range play an important role. To prove the main result of this section, we recall the following simple lemma from elementary calculus.

Lemma 2.2

Assume \(a,b,c \in {\mathbb {R}}\) and \(at^2+bt+c\) does not change sign as the real number t traverses \((-\infty ,+\infty )\). Then, the discriminant \(\varDelta :=b^2-4ac\le 0\).

The next theorem is the first which is not applicable to the case of real normal operators. Throughout the remainder of the paper, T stands for a selfadjoint operator on H.

Theorem 2.2

Let \(T=T^*\in B(H)\). Then,

$$\begin{aligned} \emptyset \ne ~\{\mathrm{extreme~ points~ of~}~ \overline{W_{\mathbb {F}}(T)}\}\cap \{-||T||,||T||\}\subset \sigma _{\mathbb {F}}(T)\subset \overline{W_{\mathbb {F}}(T)}; \end{aligned}$$
(2.2)

in particular, we show that, if \(\lambda\) is an extreme point of \(\overline{W_{\mathbb {F}}(T)}\), then

$$\begin{aligned} \lim _n(\lambda I-T)\xi _n=0~ \mathrm{~whenever~}~\lim _n\langle T\xi _n,\xi _n\rangle =\lambda ~\mathrm{and~} ~||\xi _n||=1 (n=1,2,3,\cdots ). \end{aligned}$$
(2.3)

Moreover,

$$\begin{aligned} ||N^*N||= & {} ||N||^2~\mathrm{~whenever~}~N\in B(H),~N^*N=NN^*; \end{aligned}$$
(2.4)
$$\begin{aligned} ||f(T)||= & {} ||f||_{\sigma _{\mathbb {F}}(T)}:=\max \{|f(t)| :~t\in \sigma _{\mathbb {F}}(T)\}~\mathrm{for~all~}f\in {\mathbb {F}}[x]. \end{aligned}$$
(2.5)

Proof

The inclusion \(\sigma _{\mathbb {F}}(T)\subset \overline{W_{\mathbb {F}}(T)}\) follows from Lemma 2.1. Now, let \(\lambda \in {\mathbb {F}}\) be an extreme point of \(\overline{W_{\mathbb {F}}(T)}\) and choose a finite or infinite sequence of unit vectors \(\{\xi _n\}_n\) such that \(\lim _n\langle T\xi _n,\xi _n\rangle =\lambda\). Then, \(\lambda\) is either the supremum or the infimum of the set. Assume without loss of generality that \(\lambda =0=\sup (\overline{W_{\mathbb {F}}(T)})\) and \(W_{\mathbb {F}}(T)\subset (-\infty ,0]\). Thus, \(\{\xi _n\}_n\) is a sequence of unit vectors such that \(\lim _n\langle T\xi _n,\xi _n\rangle =0\). (Certainly, such a sequence exists.) We must show that \(\lim _nT\xi _n=0\). If not, there exists a subsequence \(\{\psi _n\}_n\) of \(\{\xi_n\}_n\) and a positive number \(\epsilon\) such that \(||T\psi _n||\ge \epsilon\) for all \(n\in {\mathbb {N}}\). Then

$$\begin{aligned} 0\ge & {} \limsup _n\langle T(\psi _n+tT\psi _n),\psi _n+tT\psi _n\rangle \nonumber \\= & {} \limsup _n(t^2\langle T^2\psi _n,T\psi _n\rangle +2 t||T\psi _n||^2). \end{aligned}$$

Since the sequences \(\{\langle T^2\psi _n,T\psi _n\rangle \}_n\) and \(\{2 ||T\psi _n||^2\}_n\) are bounded, one can further reduce to a subsequence to assume without loss of generality that \(\lim _n\langle T^2\psi _n,\) \(T\psi _n\rangle =a\) and by a repeated such reduction that \(\lim _n~2 ||T\psi _n||^2=b\ge \epsilon ^2\) for some \(a,b\in {\mathbb {R}}.\) Thus, \(at^2+bt\le 0\) on \({\mathbb {R}}\) and, with the notation of Lemma 2.2, \(0<\epsilon ^2\le b^2=b^2-4ac=\varDelta \le 0\); a contradiction. Therefore, \(\lim _n||T\xi _n||=0\) and \(0\in \sigma _{\mathbb {F}}(T)\). Going back to a general extreme point \(\lambda\), it follows that \(\sigma _{\mathbb {F}}(T)\) contains the infimum and the supremum of \(W_{\mathbb {F}}(T)\) and, hence, the spectrum of T is nonempty. Consequently, \(||T||^2\ge ||T^2||\ge \sup W_{\mathbb {F}}(T^2)\in \sigma _{\mathbb {F}}(T^2)\). (Note that \(W_{\mathbb {F}}(T^2)\subset [0,||T^2||]\).)

To complete the proof of (2.2)–(2.3), it remains to show that \(\sigma _{\mathbb {F}}(T)\cap \{-||T||,\) \(||T||\}\ne \emptyset\). Let \(x_n\) be a sequence of unit vectors such that \(||Tx_n||\rightarrow ||T||\) as \(n\rightarrow \infty.\) Then \(\langle T^2x_n,x_n\rangle =||Tx_n||^2\rightarrow ||T||^2\) as \(n\rightarrow \infty\) which implies that \(||T||^2=||T^2||=\sup W_{\mathbb {F}}(T^2)=\max \sigma _{\mathbb {F}}(T^2)\). Now, Theorem 2.1 implies that \(||T||^2\in (\sigma _{\mathbb {F}}(T))^2\) and, hence, \(\sigma _{\mathbb {F}}(T)\) contains ||T|| and/or \(-||T||\).

For (2.4), note that \(N^*N\) is a (real or complex) selfadjoint operator and, hence,

$$\begin{aligned} ||N||^2=\sup _{||x||=1}||Nx||^2=\sup _{||x||=1}\langle N^*Nx,x\rangle =||N^*N||. \end{aligned}$$

Finally, to prove (2.5), let \(f(x)\equiv \sum _{j=0}^n(a_j+ib_j)x^j\) and define \(u(x)=\sum _{j=0}^na_jx^j\) and \(v(x)=\sum _{j=0}^nb_jx^j\). Then, for all \(t\in \sigma _{\mathbb {F}}(T)(\subset {\mathbb {R}})\), \(f(t)=u(t)+iv(t)\), \(|f(t)|^2=u(t)^2+v(t)^2\) and u, v are polynomials with real coefficients. It follows that

$$\begin{aligned} ||f(T)||^2= & {} ||[f(T)]^*f(T)||=||(u^2+v^2)(T)||=\sup \{|s|:~s\in \sigma _{\mathbb {F}}((u^2+v^2)(T))\}\\= & {} \sup \{|s|:~s\in (u^2+v^2)(\sigma _{\mathbb {F}}(T))\}=\sup \{u^2(t)+v^2(t):~t\in \sigma _{\mathbb {F}}(T)\}\\= & {} \sup \{|f(t)|^2:~t\in \sigma _{\mathbb {F}}(T)\}=||f||^2_{\sigma _{\mathbb {F}}(T)}. \end{aligned}$$

\(\square\)

Corollary 2.1

Let \(T=T^*\in B(H)\). The following assertions are true.

  1. (a)

    \(\min \sigma _{\mathbb {F}}(T)=\inf W_{\mathbb {F}}(T)\le \sup W_{\mathbb {F}}(T)=\max \sigma _{\mathbb {F}}(T)\).

  2. (b)

    If an extreme point of \(\overline{W_{\mathbb {F}}(T)}\) belongs to \(W_{\mathbb {F}}(T)\), then it is an eigenvalue of T.

  3. (c)

    If \(\sigma _{\mathbb {F}}(T)\subset \{\lambda \}\), then \(T=\lambda I\).

Proof

Since \(\inf ( W_{\mathbb {F}}(T))\) and \(\sup ( W_{\mathbb {F}}(T))\) are the extreme points of \(\overline{ W_{\mathbb {F}}(T)}\), it follows from (2.2) that \(\min (\sigma _{\mathbb {F}}(T))\le \inf ( W_{\mathbb {F}}(T))\le \sup ( W_{\mathbb {F}}(T))\le \max (\sigma _{\mathbb {F}}(T))\); the desired equalities are immediate from (2.2). This proves (a); the proof of (b) follows from (2.3).

For (c), if \(\sigma _{\mathbb {F}}(T)=\emptyset\), then \(H=\{0\}\) and \(S=0=\lambda I\); otherwise, \(W_{\mathbb {F}}(T)=\{\lambda \}\). Let \(K={\mathcal {K}}(\lambda I-T)\) and write \(H=K\oplus K^\perp\). Observe that \(\sigma _{\mathbb {F}}(T|_{K^\perp })\subset \{\lambda \}\) and that \(\lambda\) is not an eigenvalue of \(T|_{K^\perp }\). Thus, \({\mathcal {K}}^\perp =\{0\}\).\(\square\)

The following example shows that the spectral theory for real nonselfadjoint normal operators cannot go beyond Theorem 2.2.

Example 2.1

Consider the normal (unitary) operator \(N=U_1\oplus U_2\in B({\mathbb {R}}\oplus {\mathbb {R}}^2)\) with the following matrix representations:

$$\begin{aligned} U_1=[1];~\mathrm{and~}~U_2=\begin{bmatrix}0&{}1\\ -1&{}0\end{bmatrix}. \end{aligned}$$

Then for \(x=[x_1,x_2,x_3]^T\in {\mathbb {R}}^3\) with \(x_1^2+x_2^2+x_3^2=1\), we have \(\langle Nx,x\rangle =x_1^2\) and, hence, \(\sigma _{\mathbb {R}}(N)=\{1\}\subset W_{\mathbb {R}}(N)=[0,1]\). However, the extreme point 0 of \(W_{\mathbb {R}}(N)\) is not an eigenvalue of the (unitary) operator N. Also, note that \(\max _{t\in \{1\}}|t-1|=0<1=||N-I||\). In fact, the \(2\times 2\) unitary part \(U_2\) of N has an empty spectrum. Also, if \(f(t)\equiv t^2\), then \(f(U_2)=U_2^2=-I\) and \(\sigma _{\mathbb {R}}(f(U_2))=\{-1\}\ne \emptyset =f(\sigma _{\mathbb {R}}(U_2))\).

3 Borel functional calculus for normal operators

In this section, we study the Borel functional calculus of normal operators of the form \(N=T+i\tau\), where \((T, \tau )\) is a commuting pair of selfadjoint operators on a Hilbert space H over the field \({\mathbb {F}}\). In case \(\tau \ne 0\), the field \({\mathbb {F}}\) is necessarily equal to \({\mathbb {C}}\); otherwise, \({\mathbb {F}}\) can be either \({\mathbb {C}}\) or \({\mathbb {R}}\). For a vector \(x\in H\) and a normal operator \(N=T+i\tau \in B(H)\), the smallest invariant subspace and the smallest reducing invariant subspace of N containing x can be, respectively, formulated as follows:

$$\begin{aligned} {\mathcal {Z}}(N;x)= & {} {\bar{\vee }}\{N^mx:~m=0,1,2,\cdots \}~\mathrm{~and}\\ {\mathcal {Z}}(N,N^*;x):= & {} {\mathcal {Z}}(T,\tau ;x)={\bar{\vee }}\{N^mN^{*n}x:~m,n=0,1,2,\cdots \}\\= & {} {\bar{\vee }}\{T^m\tau ^{*n}x:~m,n=0,1,2,\cdots \}; \end{aligned}$$

i.e., the closure of the linear span of all \(p(T,\tau )x\) as p runs in the collection of all bivariate polynomials with coefficients in \({\mathbb {F}}\). In case \(\tau =0\), we have \({\mathcal {Z}}(N,N^*;x)={\mathcal {Z}}(T;x)\). Note that every closed invariant subspace K of \(\{T,\tau \}\) is a reducing one; i.e., \(N=N_1\oplus N_2\) with respect to the direct sum \(H=K\oplus K^\perp\), where

$$\begin{aligned} \sigma _{\mathbb {F}}(N)=\sigma _{\mathbb {F}}(N_1)\cup \sigma _{\mathbb {F}}( N_2)~\mathrm{and~}~N_\gamma N_\gamma ^*=N_\gamma ^*N_\gamma ~(\gamma =1,2). \end{aligned}$$

The following lemma is a consequence of the results of the previous section; its simple proof is left to the interested reader.

Lemma 3.1

(Continuous functional calculus for selfadjoint operators) Let \(T=T^*\in B(H)\) and let \(\mathbf{1}\) and \(\mathbf{id}\) be the polynomials \(p(t)\equiv 1\) and \(p(t)\equiv t\), respectively. Then, the polynomial functional calculus \(p\mapsto p(T):{\mathbb {F}}[x]\rightarrow B(H)\) is an isometric \(^*\)-algebra isomorphism such that \(\mathbf{1}(T)=I\), \(\mathbf{id}(T)=T\). Moreover, the functional calculus can be extended to an isometric \(^*\)-algebra isomorphism \(f\mapsto f(T):C_{\mathbb {F}}(\sigma _{\mathbb {F}}(T))\rightarrow B(H)\) such that

$$\begin{aligned} ||f(T)||=||f||_{\sigma _{\mathbb {F}}(T)}, ~\forall f\in C_{\mathbb {F}}(\sigma _{\mathbb {F}}(T)). \end{aligned}$$

To prepare ourselves for the main theorem, we need the following definition.

Definition 3.1

A function \(f:{\mathbb {R}}^2\rightarrow {\mathbb {C}}\) may be also regarded as a function \({\tilde{f}}:{\mathbb {C}}\rightarrow {\mathbb {C}}\) by \({\tilde{f}}(z)=f(s(z),t(z))\), where \(s(z)=(z+z^*)/2\) and \(t(z)=(z-z*)/(2i)\). Now, if \(N=T+i\tau\) is a normal operator, then we may write \({\tilde{f}}(N)\) to mean \(f(T,\tau )\) provided that the latter expression is somehow defined. If no ambiguity arises, we may even drop the tilde sign to write f(N) instead of \(f(T,\tau )\).

Theorem 3.1

Let \(N\in B(H)\) be a real selfadjoint or a complex normal operator. Then there exist a positive Borel measure \(({\mathbb {F}},{\mathfrak {B}},\mu _x^N)\) and a unitary operator \(U_N:L^2_{\mathbb {F}}(\mu _x^N)\rightarrow {\mathcal {Z}}(N,N^*;x)\) such that for all \(f\in {\mathbb {F}}[x]\), for all \(\phi \in L^2_{\mathbb {F}}(\mu _x^N)\) and for all \(y\in H\) ,

$$\begin{aligned}&U_{T,\tau ;x}f=f(N)x,~~||\mu _x^N||=\mu _x^N(\sigma _{\mathbb {F}}(N))=||x||^2,~ \mathrm{~and} \end{aligned}$$
(3.1)
$$\begin{aligned}&\langle U_{T,\tau ;x}\phi ,y\rangle =\int \phi (s)\{[U_{T,\tau ;x}^*Py](s)\}^*d\mu _x^N(s), \end{aligned}$$
(3.2)

where \(P:H\rightarrow H\) is the orthogonal projection onto \({\mathcal {Z}}(N,N^*;x)\). Moreover, if \({\mathcal {A}}_b(S)\) denotes the \(^*\)-algebra of all \({\mathbb {F}}\)-valued Borel functions bounded on \(\sigma _{\mathbb {F}}(S)\), then the mapping \(f\mapsto f(N):{\mathcal {A}}_b(N) \rightarrow B(H)\) defined by \(f(N) x=U_{T,\tau ;x}f\) is a \(^*\)-algebra homomorphism extending the continuous functional calculus of Lemma 3.1. Moreover,

$$\begin{aligned} Qf(N)=f(N)Q,~\mathrm{~whenever~}~QN=NQ,~\mathrm{for~some~}~Q\in B(H). \end{aligned}$$
(3.3)

Proof

Assume for the moment that \(\tau =0\) and, hence, \({\mathbb {F}}={\mathbb {R}}\). In view of Lemma 3.1, for each \(x\in H\), the mapping \(f\mapsto \langle f(T)x,x\rangle :C_{\mathbb {R}}(\sigma _{\mathbb {R}}(T))\rightarrow {\mathbb {R}}\) defines a positive linear functional \(\Phi _x\) such that \(|\Phi _x(f)|=|\langle f(T)x,x\rangle |\le ||f(T)||~||x||^2=||f||_{\sigma _{\mathbb {R}}(T)}~||x||^2\). Letting \(f=\mathbf{1}\) yields \(|\langle f(T)x,x\rangle |=||x||^2\) and, thus, \(||\Phi _x||=||x||^2\). Hence, there exists a unique positive measure \((\sigma _{\mathbb {R}}(T),{\mathfrak {B}},\mu _x^T)\) such that \(||\mu ^T_x||=||x||^2\) and \(\langle f(T)x,x\rangle =\int fd\mu ^T_x\) for all \(f\in C_{\mathbb {R}}(\sigma _{\mathbb {R}}(T))\). Also, note that

$$\begin{aligned} ||f(T)x||^2= & {} \langle f(T)x,f(T)x\rangle =\langle f^*(T)f(T)x,x\rangle =\langle |f|^2 (T)x, x \rangle \nonumber \\= & {} \int |f(s)|^2d\mu _x^T(s)=||f||^2_{L^2(\mu _x)},~\forall ~f\in \sigma _{\mathbb {R}}(T). \end{aligned}$$
(3.4)

The equation (3.4) defines an isometry \(U_0\) from a dense subset of \(L^2_{\mathbb {R}}(\mu _x^T)\) onto a dense subset of \({\mathcal {Z}}(T;x)\) which can be extended to a unitary operator \(U_{T;x}\) from \(L^2_{\mathbb {R}}(\mu _x^T)\) onto \({\mathcal {Z}}(T;x)\) satisfying (3.1). The proof of (3.2) follows from the fact that \(\phi\), \(~U_{T;x}^*Py\in L^2_{\mathbb {R}}(\mu _x^T)\) and

$$\begin{aligned} \langle U_{T;x}\phi ,y\rangle =\langle U_{T;x}\phi ,Py\rangle =\langle \phi ,U_{T;x}^*Py\rangle =\int \phi (s)\{[U_{T;x}^*Py](s)\}^* d\mu _x^T(s). \end{aligned}$$

Next, let \(f\in L^\infty _{\mathbb {R}}(\mu _x^T)\) and define \(M_f:L^2_{\mathbb {R}}(\mu _x^T)\rightarrow L^2(\mu _x^T)\) by \(M_f\phi =f\phi\). It is well known that \(M_f\) is a bounded normal operator of norm \(||M_f||=||f||_\infty\) with adjoint \(M_f^*=M_{f^*}\). This makes us ready to define the following Borel functional calculus for any real or complex selfadjoint operator T:

$$\begin{aligned} f\mapsto f(T):{\mathcal {A}}_b(T)\rightarrow B(H)~\mathrm{~via~}~f(T)x = U_{T;x}f,~\forall ~ x\in H. \end{aligned}$$
(3.5)

We show that f(T) is a well-defined bounded linear operator defined on H which satisfies (3.3), \(||f(T)||\le ||f||_{\sigma _{\mathbb {R}}(T)}\), \(f(T)g(T)=(fg)(T)\), and \(f(T)^*=f^*(T)\) \((\forall f,g\in {\mathcal {A}}_b(T))\). For arbitrary \(x,y\in H\) and \(\alpha \in {\mathbb {R}}\), define \(\omega =\mu _x^T+\mu _y^T+\mu _{\alpha x+y}^T+\mu _{Qx}^T+\mu _{g(T)x}^T\) and choose sequences \(\{f_n\}\) and \(\{g_n\}\) of continuous functions converging to f and g in \(L^2_{\mathbb {R}}(\omega )\), respectively. We modify \(f_n\) and \(g_n\) to assume without loss of generality that \(||f_n||_{\sigma _{\mathbb {R}}(T)}\le ||f||_{\sigma _{\mathbb {R}}(T)}+1\) and \(||g_n||_{\sigma _{\mathbb {R}}(T)}\le ||g||_{\sigma _{\mathbb {R}}(T)}+1\) for all \(n\in {\mathbb {N}}\). Hence, in view of the bounded convergence theorem, \(f_ng_n\) converges to fg in \(L^2_{\mathbb {R}}(\omega )\). (Note that, \(\omega\) is supported on \(\sigma _{\mathbb {R}}(T)\subset {\mathbb {R}}\).) Then, for \(u=x,y,\alpha x+y\), Qx, and g(T)x,

$$\begin{aligned} f(T)u=U_{T,u}f=\lim _nU_{T,u}f_n=\lim _nf_n(T)u. \end{aligned}$$
(3.6)

Therefore, the following assertions are true.

  1. (a)

    \(f(T)(\alpha x+y)=\lim _nf_n(T)(\alpha x+y)=\alpha \lim _nf_n(T)x+\lim _nf_n(T)y=\alpha f(T)x+f(T)y\) which implies that f(T) is a well-defined linear operator.

  2. (b)

    In view of (3.6) and the commutativity property \(Qf_n(T)=f_n(T)Q\), it follows that \(Qf(T)x-f(T)Qx=\lim _n[Qf_n(T)x-f_n(T)Qx]=0\), which proves (3.3).

  3. (c)

    In view of (3.4),

    $$\begin{aligned} ||f(T)x||^2= & {} \lim _n||f_n(T)x||^2=\lim _n\int |f_n|^2d\mu _x^T \le \lim _n||f_n||^2_{\sigma _{\mathbb {R}}(T)} ||x||^2\\\le & {} ||f||^2_{\sigma _{\mathbb {R}}(T)}||x||^2, \end{aligned}$$

    which implies that f(T) is bounded.

  4. (d)

    \(\langle f(T)x,y\rangle =\lim _n \langle f_n(T)x,y\rangle =\lim _n \langle x,f_n(T)^*y\rangle = \lim _n \langle x,f_n^*(T)y\rangle = \langle x,\) \(f^*(T)y\rangle\) which shows that \(f(T)^*=f^*(T)\).

  5. (e)

    Finally,

    $$\begin{aligned} ||\{(fg)(T)-f(T)g(T)\}x||= & {} \lim _n||\{(f_ng_n)(T)x-f_n(T)(g(T)x)||\\= & {} \lim _n||f_n(T)\{g_n(T)x-g(T)x\}||\\\le & {} \lim _n||f_n(T)||~||[g_n(T)-g(T)]x||\\\le & {} (||f||_{\sigma _{\mathbb {F}}(T)}+1) \lim _n[\int |g_n-g|^2d\mu _x^T]^{1/2}=0, \end{aligned}$$

    which implies that \((fg)(T)=f(T)g(T)=g(T)f(T)\) and that f(T) is normal.

This completes the proof of the theorem for the case \(N=T\).

We now assume \(N=T+i\tau\) with \(\tau \ne 0\); here, the underlying field \({\mathbb {F}}\) is necessarily equal to \({\mathbb {C}}\). Choose a fixed rectangle \(\varGamma =[{\bar{a}},{\bar{b}}]\times [{\bar{c}},{\bar{d}}]\) such that \(\sigma _{\mathbb {F}}(N)\subset \varGamma\). Let \({\mathfrak {S}}\) be the algebra generated by all sets of the form \([a,b)\times [c,d)\cap \varGamma\). Define \(\mu _x^N:{\mathfrak {S}}\rightarrow [0,\infty ]\) as follows:

$$\begin{aligned} \mu _x^N([a,b)\times [c,d))= & {} \langle \chi _{[a, b)}(T)x,\chi _{[c,d)}(\tau )x\rangle \nonumber \\= & {} \int \chi _{[a, b)}(s)\{[U_{T;x}^*P\chi _{[c, d)}(\tau )x](s)\}^*d\mu _x^T(s), \end{aligned}$$
(3.7)

where \(P:H\rightarrow H\) is the orthogonal projection onto \({\mathcal {Z}}(T;x)\). To extend \(\mu ^N_x\) to a Borel \(\sigma\)-additive measure, we have to show that it is \(\sigma\)-additive on the ring \({\mathfrak {S}}\). Let \([a,b)\times [c,d)= {\dot{\cup }}_n([a_n,b_n)\times [c_n,d_n))\). Then

$$\begin{aligned}&\mu ^N_x([a,b)\times [c,d))=\langle \chi _{[a,b)}(T)x,\chi _{[c,d)}(\tau )x\rangle \nonumber \\ & = \int \chi _{[a, b)}(s)\{[U_{\tau ;x}^*P\chi _{[c, d)}(\tau )x](s)\}^*d\mu _x^T(s)\nonumber \\& = \int \left( \int _c^d\sum _{n=1}^\infty \chi _{[a_n, b_n)}(s)\frac{\chi _{[c_n,d_n)}(t)}{d_n-c_n} \{[U_{\tau ;x}^*P\chi _{[c_n, d_n)}(\tau )x](s)\}^*dt\right) d\mu _x^T(s) \nonumber \\& = \sum _{n=1}^\infty \int \int _c^d\chi _{[a_n, b_n)}(s)\frac{\chi _{[c_n,d_n)}(t)}{d_n-c_n}\{[U_{\tau ;x}^*P\chi _{[c_n, d_n)}(\tau )x](s)\}^*dtd\mu _x^T(s) \nonumber \\& = \sum _{n=1}^\infty \int \chi _{[a_n, b_n)}(s)\{[U_{\tau ;x}^*P\chi _{[c_n, d_n)}(\tau )x](s)\}^*d\mu _x^T(s) \nonumber \\& = \sum _{n=1}^\infty \mu ^N_x([a_n,b_n)\times [c_n,d_n)). \end{aligned}$$
(3.8)

Thus, \(\mu _x^N\) is a Borel positive measure on \(\varGamma\) and can be extended to a measure space \(({\mathbb {C}},{\mathfrak {B}}, \mu _x^N)\) by defining \(\mu _x^N(B)=\mu _x^N(B\cap \varGamma )\) for all \(B\in {\mathfrak {B}}\).

Thus, for simple Borel functions \(f=\sum _{i=1}^mc_i\chi _{_{E_i}}\) and \(g=\sum _{j=1}^nd_j\chi _{_{F_j}}\),

$$\begin{aligned} \langle f(T)g(\tau )x,x\rangle =\langle f(T)x,g^*(\tau )x\rangle = \int f(s)g(t)d\mu _x^N(s,t). \end{aligned}$$
(3.9)

It follows easily that (3.9) as well as the following equation hold for all bivariate polynomials f(s, t) with coefficients in \({\mathbb {C}}\):

$$\begin{aligned} ||f(T,\tau )x||^2=\langle |f|^2(T,\tau )x,x\rangle =\int |f(s,t)|^2d\mu ^N_x(s,t). \end{aligned}$$
(3.10)

Thus, in view of the Stone–Weierstrass Theorem, \(~{\mathcal {Z}}(N,N^*;\,x)=U_{T,\tau ;\,x}L^2_{\mathbb {C}}(\mu _x^N)\) for some unitary operator \(U_{T,\tau ;\,x}:L^2_{\mathbb {C}}(\mu _x^N)\rightarrow {\mathcal {Z}}(N,N^*;\,x)\) satisfying \(U_{T,\tau ;\,x}f = f(T,\tau )x\) for all functions of the form \(f(s,t)=\sum _{j,k=1}^nc_{jk}g_j(s)h_k(t)\) with \(g_j\in C_{\mathbb {C}}([{\bar{a}},{\bar{b}}])\), \(h_k\in C_{\mathbb {C}}([{\bar{c}},{\bar{d}}])\).

We now replace T by \(N=T+i\tau\), \(\mu _x^T\) by \(\mu _x^N\), and U(T; x) by \(U(T,\tau ;\,x)\), and then mimic the arguments leading to (3.5)–(3.6) to complete the proof of the theorem in the general case.\(\square\)

We now try to equip \({\mathcal {A}}_b(N)\) with a norm with respect to which the functional calculus \(f\mapsto f(T,\tau ):{\mathcal {A}}_b(N)\rightarrow B(H)\) is an isometric \(^*\)-algebra isomorphism. Note that \(\mathcal A_b(N)\subset \cap _{x\in H}L^\infty (\mu _x^N)\) and this will help us in the norm construction. Let us first fix a notation to be used in the theorem.

Definition 3.2

Let \(({\mathbb {F}},{\mathfrak {B}},\mu )\) be a positive Borel measure and let \(f:{\mathbb {F}}\rightarrow {\mathbb {F}}\) be measurable. The \([\mu ]\)essential range of f, denoted by \([\mu ]\)ess.range(f), is the set of all \(\zeta \in {\mathbb {F}}\) for which \(\mu (f^{-1}(\varDelta ))>0\) for every neighborhood \(\varDelta\) of \(\zeta\). Also, supp\((\mu )=\cap \{C={\bar{C}}:~\mu ({\mathbb {F}}\backslash C)=0\}\).

Remark 3.1

It is easy to see that, for every \(\mu\)-measurable set \(\gamma\) and every \(\mu\)-measurable function \(f:{\mathbb {F}}\rightarrow {\mathbb {F}}\),

$$\begin{aligned}&\mathrm{supp}(\mu |_\gamma )\subset [\mu ]\mathrm{ess.range}(\mathbf{id~}\cdot \chi _\gamma )\subset \mathrm{supp}(\mu |_\gamma )\cup \{0\}, \end{aligned}$$
(3.11)
$$\begin{aligned}&[\mu ]\mathrm{ess.range}(f)=\cap \{\overline{g(\mathrm{supp}(\mu ))}:~g=f~a.e.[\mu ]\}, \end{aligned}$$
(3.12)

where \((\mathbf{id}\cdot \chi _\gamma )(t)\equiv t\chi _\gamma (t)\).

The following theorem is an improvement of Theorems 4.6, 4.8, 4.11 of [7] and of Theorem 5.5.3 of [13].

Theorem 3.2

Suppose the notation and the hypotheses of Theorem 3.1 are valid. For each \(f\in {\mathcal {A}}_b(N)\), define \(\nu (f)=\sup _{x\in H}||f||_{L^\infty (\mu _x^N)}\). Then, \(\nu\) is a seminorm on \({\mathcal {A}}_b(N)\) which induces a norm \(||\cdot ||_N\) on the completion \(L^\infty (N)\) of the quotient space \({\mathcal {A}}_b(N)/\{f:\nu (f)=0\}\) with respect to which the functional calculus \(f\mapsto f(T,\tau ):L^\infty (N)\rightarrow B(H)\) is an isometric \(^*\)-algebra isomorphism. Moreover, the following assertions are true for all \(f,g\in L^\infty (N)\).

  1. (i)

    \(\sigma _{\mathbb {F}}(f(T,\tau ))=\varLambda\), where \(\varLambda :=\cup _{x\in H}~[\mu _x^N]\mathrm{ess.range}(f)\).

  2. (ii)

    \(\sigma _{\mathbb {F}}(N)=\sigma _{\mathbb {F}}(T+i\tau )=\cup _{x\in H}~\mathrm{supp}(\mu _x^N)\) and \(\sigma _{\mathbb {F}}(f(T,\tau ))\subset \overline{f(\sigma _{\mathbb {F}}(N))}\).

  3. (iii)

    \(f(T,\tau )=g(T,\tau )\) whenever \(f(s,t)=g(s,t)\) for all \((s,t)\in \sigma _{\mathbb {F}}(N)\).

  4. (iv)

    \(||f||_N=||f||_{\sigma _{\mathbb {F}}(N)}:=\max \{|f(t)|:~t\in \sigma _{\mathbb {F}}(N)\}\) if \(f|_{\sigma _{\mathbb {F}}(N)}\) is continuous.

Proof

Write \(||f||_x\) as a short form for \(||f||_{L^\infty (\mu _x^N)}\). It is clear that \(\nu (f)\ge 0\), \(\nu (cf)=\sup _x||cf||_x=|c|\sup _x||f||_x=|c|\nu (f)\), and

$$\begin{aligned} \nu (f+g)=\sup _x||f+g||_x\le \sup _x||f||_x+\sup _x||g||_x =\nu (f)+\nu (g), \end{aligned}$$

for all \(c\in {\mathbb {F}}\), for all \(x\in H\) and for all \(f,g\in L^\infty (N)\). Thus \(\nu\) is a seminorm and induces a norm \(||\cdot ||_N\) on \(L^\infty (N)\). Also, with the same notation, it follows from (3.10) that

$$\begin{aligned} ||f(T,\tau )x||^2=||U_{T,\tau ;\,x}f||^2=\int |f|^2d\mu _x^N\le ||f||_x^2||x||^2 \end{aligned}$$

which implies that \(||f(T,\tau )||\le ||f||_{_N}\). Next, choose \(x\in H\) such that \(||f||_{L^\infty (\mu _x)}>||f||_{_N}-\epsilon /2\) for a given \(\epsilon >0\). Let \(\varDelta\) be a Borel set of measure \(m=\mu _x^N(\varDelta )>0\) on which \(|f(t)|>||f||_{L^\infty (\mu _x)}-\epsilon /2>||f||_{_N}-\epsilon\), a.e.\([\mu _x^N]\). Define \(g=m^{-1/2}\chi _{_\varDelta }\). Then

$$\begin{aligned} ||g(T,\tau )x||^2=\int |g(s,t)|^2d\mu _x^N(s,t)=1, \end{aligned}$$

and

$$\begin{aligned} ||f(T,\tau )g(T,\tau )x||^2= & {} \int |f(s,t)g(s,t)|^2d\mu _x^N(s,t)\\\ge & {} (||f||_{_N}-\epsilon )^2\int |g(s,t)|^2d\mu _x^N(s,t)=(||f||_{_N}-\epsilon )^2. \end{aligned}$$

Thus, \(||f(T,\tau )||\ge ||f||_{_N}-\epsilon\) and, since \(\epsilon >0\) is arbitrary, it follows that \(||f(T,\tau )||= ||f||_{_N}\). It is now easy to verify that the functional calculus \(f\mapsto f(T,\tau ):L^\infty (N)\rightarrow B(H)\) is an isometric \(^*\)-algebra isomorphism.

To prove (i), we first show that \(\varLambda\) is closed. By (3.12), the set \(\varLambda _x:=[\mu _x^N]\mathrm{ess.range}(f)\) is closed for all \(x\in H\). Fix \(\lambda \in {\overline{\varLambda }}\). Assume no \(\varLambda _x\) contains \(\lambda\) and reach a contradiction. Let \(D_1\) be the closed disc/interval in \({\mathbb {F}}\) of radius \(r_1= 1\) centered at \(\lambda\). Find \(x\in H\) such that \(D_1\cap \varLambda _x\ne \emptyset\). We are going to replace x by a unit vector \(x_1\) to guarantee (3.13) holds for \(n=1\). (See below.) Define \(\varGamma _1=f^{-1}( D_1)\) and let \(x_1=U_{T,\tau ;\,x}g_1\), where \(g_1=\mu _x^N(\varGamma _1)^{-1/2}\chi _{_{\varGamma _1}}\). Then, for all Borel subsets \(\Omega\) of \({\mathbb {F}}\),

$$\begin{aligned} \mu _{x_1}^N(\Omega )= & {} \int |\chi _{_\Omega }(s,t)|^2d\mu _{x_1}^N=||\chi _{_\Omega }(T,\tau )x_1||^2=||\chi _{_\Omega }(T,\tau )U_{T,\tau ;\,x}\chi _{_{\varGamma _1}}||^2/\mu _x^N(\varGamma _1)\\= & {} ||\chi _{_{\Omega \cap \varGamma _1}}||^2_{L^2(\mu _x^N)}/\mu _x^N(\varGamma _1)=\mu _x^N(\Omega \cap \varGamma _1)/\mu _x^N(\varGamma _1), \end{aligned}$$

which implies that \(||x_1||^2=\mu _{x_1}({\mathbb {F}})=1\), and \(\varLambda _{x_1}=\varLambda _x\cap D_1\ne \emptyset\). Since \(\varLambda _{x_1}\) is closed and \(\lambda \notin \varLambda _{x_1}\), it follows that \(\varLambda _{x_1}\subset D_1\backslash D_2\) for some closed disc of (positive) radius \(r_2\le r_1/2\) and centered at \(\lambda\). Now, assume by induction, we have constructed a sequence of concentric closed discs \(D_1,D_2,D_3, \ldots ,D_{n+1}\) of respective radii \(r_1,r_2,\ldots ,r_k\) and an orthonormal set \(\{x_1,x_2, x_3,\ldots , x_n\}\subset H\) such that, for \(k=1,2,\ldots ,n\),

$$\begin{aligned} r_{k+1}\le r_k/2,{~}~ x_k=\chi _{_{\varGamma _k\backslash \varGamma _{k+1}}}(T,\tau )x_k~\mathrm{~and~}~\varLambda _{x_k}\subset D_k\backslash D_{k+1}, \end{aligned}$$
(3.13)

where \(\varGamma _k=f^{-1}(D_k)\). Mimicking the construction of \(x_1\) and \(D_2\), yields a unit vector \(x_{n+1}\in H\) and a disc \(D_{n+2}\) of radius \(r_{n+1}\le r_n/2\) centered at \(\lambda\) such that (3.13) remains valid for \(k=n+1\). We claim \(\langle x_k,x_{n+1}\rangle =0\) for \(k=1,2,\ldots ,n\). In view of (3.13 )

$$\begin{aligned} \langle x_k,x_{n+1}\rangle= & {} \langle \chi _{_{\varGamma _k\backslash \varGamma _{k+1}}}(T,\tau )x_k,\chi _{_{\varGamma _{n+1}\backslash \varGamma _{n+2}}}(T,\tau )\rangle \\= & {} \langle \chi _{_{\varGamma _{n+1}\backslash \varGamma _{n+2}}}(T,\tau )\chi _{_{\varGamma _k\backslash \varGamma _{k+1}}}(T,\tau )x_k,x_{n+2}\rangle =0,{~}~{~}\forall k\le n. \end{aligned}$$

So far, we have constructed an orthonormal sequence \(x_1,x_2,x_3,\ldots\) such that \(\mu _{x_k}^N(f^{-1}(D_k))\ne 0\). Define \(x=\sum _{k=1}^\infty 2^{-k}x_k\) and observe that

$$\begin{aligned} ||x||=1,~{~}~\mathrm{and~}~\mu _x^N(f^{-1}(D_k\backslash D_{k+1})=\mu _x^N(\varGamma _k\backslash \varGamma _{k+1})>0. \end{aligned}$$

Thus, \(D_k\backslash D_{k+1}\subset [\mu _x^N]\mathrm{ess.range}(f)\). This implies that \(\lambda \in {\overline{\varLambda }}_x=\varLambda _x\).

It remains to show that \(\lambda \in \sigma _{\mathbb {F}}(f(T,\tau ))\). Then there exists \(x\in H\) such that \(\lambda \in [\mu _x]\mathrm{ess.range}(f)\). For every \(n\in {\mathbb {N}}\), let \(\varDelta _n\) be the disc in \({\mathbb {F}}\) of radius \(n^{-1}\) and center \(\lambda\). Then, for all \(n\in {\mathbb {N}}\), \(\mu _x(\varGamma _n)>0\), where \(\varGamma _n:=f^{-1}(\varDelta _n)\). Let \(g_n=\mu _x (\varGamma _n)^{-1/2} \chi _{_{\varGamma _n}}\) and \(y_n=U_{T,\tau ;\,x}g_n\) for all \(n\in {\mathbb {N}}\). Then \(||y_n||=||g_n||_{L^2(\mu _x)}=1\) and

$$\begin{aligned} ||(\lambda -f(T,\tau ))y_n||^2=\mu _x(\varGamma _n)^{-1}\int _{\varGamma _n} |\lambda -f(s+it)|^2 d\mu _x(s,t)\le n^{-2}\rightarrow 0 \end{aligned}$$
(3.14)

Conversely, we assume \(\lambda \in \sigma _{\mathbb {F}}(f(T,\tau ))\) and claim \(\lambda \in \varLambda\). It is sufficient to show that \(\lambda \in {\overline{\varLambda }}\). If not, there exists an open disc \(\varDelta\) of radius \(\delta >0\) and center \(\lambda\) such that \(\mu _x^N(f^{-1}(\varDelta ))=0\) for all \(x\in H\). Then,

$$\begin{aligned} ||(\lambda -f(T,\tau ))x||^2=\int |\lambda -f(s+it)|^2d\mu _x^N(s,t)\ge \delta ^2 ||x||^2. \end{aligned}$$

This shows that \(\lambda\) is not in the approximate point spectrum of the normal operator \(f(T,\tau )\). Hence, \(\lambda \notin \sigma _{\mathbb {F}}(f(T,\tau ))\); a contradiction.

The proof of (ii) follows from the fact that \([\mu _x^N]\mathrm{ess.range}(\mathbf{id})=\mathrm{supp}(\mu _x^N)\) and that \(N=T+i\tau =\mathbf{id}(T,\tau )\). Thus, \(\sigma _{\mathbb {F}}(N)=\cup _{x\in H}\mathrm{supp}(\mu _x^N)\). The remainder of the proof of (ii) follows from Part (i) and formulas (3.11)–(3.12).

For (iii), observe that \(\mathrm{supp}(\mu _x^N)\subset \sigma _{\mathbb {F}}(N)\). Letting \(h=f-g\), it follows that

$$\begin{aligned} ||h(T+i\tau )x||^2= \int |h(s,t)|^2d\mu _x^N= 0,~\forall ~ x\in H. \end{aligned}$$

In (iv), since f is continuous, it follows that

$$\begin{aligned} ||f||_N=\sup _{x\in H}||f||_x\le \sup _{x\in H}\max \{|f(t)|:~t\in \sigma _{\mathbb {F}}(N)\}\le ||f||_{\sigma _{\mathbb {F}}(N)}. \end{aligned}$$

For the converse, choose \(s\in \sigma _{\mathbb {F}}(N)\) such that \(|f(s)|=\max \{|f(t)|:~t\in \sigma _{\mathbb {F}}(N)\}\). In view of Part (ii), there exists \(x\in H\) such that \(s\in \mathrm{supp}(\mu _x^N)\) and it follows from the continuity of f on the compact set supp\((\mu _x^N)\) that \(|f(s)|\le ||f||_{L^\infty (\mu _x^N)}\le ||f||_N\).\(\square\)

Corollary 3.1

Let \(\sigma _{\mathbb {F}}(N)=\sigma _1\cup \sigma _2\) for a pair of disjoint closed sets \(\sigma _1\) and \(\sigma _2\). Then \(N=N_1\oplus N_2\) with \(\sigma _{\mathbb {F}}(N_j)= \sigma _j\) \((j=1,2)\). In particular, if \(\sigma _1=\{\lambda \}\), then \(N_1=\lambda I_1\), where \(I_1=I|_{{\mathcal {D}}(N_1)}\).

Proof

Let \(\chi _j\) be the characteristic function of the set \(\sigma _j\) and define \(P_j=\chi _j(N)\) \((j=1,2)\). Obviously, each \(P_j\) is an orthogonal projection and

$$\begin{aligned}&I=\mathbf{1}(N)=\chi _1(N)+\chi _2(N)=P_1+P_2,\\&P_1P_2=P_2P_1=\chi _1(N)\chi _2(N)=(\chi _1\chi _2)(N)=0. \end{aligned}$$

Thus, \(H=H_1\oplus H_2\) and \(NH_j\subset H_j\) \((j=1,2)\). Hence, \(N=N_1\oplus N_2\), \(\sigma _{\mathbb {F}} (N)=\sigma _{\mathbb {F}}(N_1)\cup \sigma _{\mathbb {F}}( N_2)\), and \(N_1, N_2\) are normal operators, where \(N_j=N|_{H_j}\) \((j=1,2)\). But

$$\begin{aligned} \sigma (N_j)=\sigma (N\chi _j(N)|_{H_j})\subset \{t\chi _j(t):t\in \sigma _{\mathbb {F}}(N)\}\subset \sigma _j\cup \{0\},~ (j=1,2). \end{aligned}$$

Assume \(0\notin \sigma _j\) and \(x\in H_j\) for some \(j=1,2\). Then,

$$\begin{aligned} ||Nx||^2=||N\chi _j(N)x||^2=\int _{\sigma _j} |t|^2d\mu _x^N\ge \mathrm{dist}(0,\sigma _j)||x||^2, \end{aligned}$$

which implies that 0 is not in the approximate point spectrum of \(N_j\). Hence, \(\sigma _{\mathbb {F}}(N_j)\subset \sigma _j\) and, thus, \(\sigma _{\mathbb {F}}(N_j)=\sigma _j\) \((j=1,2)\).

In particular, if \(\sigma _1=\{\lambda \}\), then

$$\begin{aligned} ||(\lambda I-N)\chi _{\{\lambda \}}(N)x||^2=\int _{\{\lambda \}}|\lambda -s-it|^2d\mu _x^N = 0. \end{aligned}$$

Thus, \(N_1=\lambda I_1\).\(\square\)

We conclude this section with a simple observation about the polar decomposition of an operator \(S\in B(H)\) acting on a real (or complex) Hilbert space H. It is obvious that \(S^*S\) is a (real or complex) positive selfadjoint operator and, hence, has a positive square root \((S^*S)^{1/2}\). As usual, we denote this square root by |S|. Also, it is easy to verify that \(\langle |S|x,|S|x\rangle =\langle Sx,Sx\rangle\) which implies that the mapping \(|S|x\mapsto Sx\) is a partial isometry from the range of |S| onto the range of S satisfying \(S=V|S|\).

Now, if \(N\in B(H)\) is a real or complex normal operator, then \({\mathcal {K}}(N)={\mathcal {K}}(N^*)={\mathcal {K}}(|N|)\), \(\overline{{\mathcal {R}}(N)}=\overline{{\mathcal {R}}(N^*)}=\overline{{\mathcal {R}}(|N|)}\), \(H={\mathcal {K}}(N)\oplus \overline{{\mathcal {R}}(N)}\) and, accordingly,

$$\begin{aligned} N=\begin{bmatrix} 0&{}0\\ 0&{}N_1\end{bmatrix}=\begin{bmatrix} W&{}0\\ 0&{}V\end{bmatrix}\begin{bmatrix} 0&{}0\\ 0&{}|N_1|\end{bmatrix}, \end{aligned}$$

where \(N_1:\overline{{\mathcal {R}}(N)}\rightarrow \overline{{\mathcal {R}}(N)}\) is an injective normal operator, \(W:{\mathcal {K}}(N)\rightarrow {\mathcal {K}}(N)\) is an arbitrary operator, and \(V:\overline{{\mathcal {R}}(N)}\rightarrow \overline{{\mathcal {R}}(N)}\) is a unique unitary operator. Letting W be any unitary operator, one can define a unitary operator

$$\begin{aligned} U=\begin{bmatrix} W&{}0\\ 0&{}V\end{bmatrix}=W\oplus V \end{aligned}$$

to establish the polar decomposition \(N=U|N|\) of N with \(U^*U=UU^*=I\), regardless of the type of the underlying field \({\mathbb {R}}\) or \({\mathbb {C}}\).

At this end, assuming \(N=T+i\tau\) is a real selfadjoint or a complex normal operator, we want to demonstrate the operators U and N as functions of N. Define \(u(s,t)=(s+it)(s^2+t^2)^{-1/2}\) if \(s^2+t^2\ne 0\) and \(u(0,0)=1\). Also, define \(h(s,t)=(s^2+t^2)^{1/2}\). It follows that \(s+it=u(s,t)h(s,t)\) and, hence, \(N=u(T,\tau )h(T,\tau )\), where

$$\begin{aligned} ||u(T,\tau )x||^2= & {} \int |u|^2d\mu _x^N=||x||^2,\\ ||u(T,\tau )^*x||^2= & {} ||u^*(T,\tau )x||^2=\int |u^*|^2d\mu _x^N=||x||^2,\\ h(T,\tau )^*= & {} h^*(T,\tau )=h(T,\tau ),~\mathrm{~and}\\ \langle h(T,\tau )x,x\rangle= & {} \int hd\mu _x^N\ge 0,~{~}~\forall x\in H. \end{aligned}$$

Therefore, \(u(T,\tau )\) is a unitary operator and \(h(T,\tau )\) is a positive selfadjoint operator which together establish the polar decomposition of N. Also, note that \(u=w+v\) with \(w=\chi _{\{(0,0)\}}\), \(v=(s+it)(s^2+t^2)^{-1/2}\chi _{{\mathbb {F}}\backslash \{(0,0)\}}(s,t)\) and \(wv=0\). This re-establishes the direct sum \(U=W\oplus V\) with the unique part \(V=v(T,\tau )|_{{\mathcal {K}}(N)^\perp }\) and the choice \(W=w(T,\tau )|_{{\mathcal {K}}(N)}=I_{{\mathcal {K}}(N)}\).

Here, we make a very important observation about the relation between the spectrum of U and that of |N|; i.e., given a Borel partition \({\mathbb {T}}=\eta \cup \omega\) of the torus \({\mathbb {T}}:=\{z\in {\mathbb {F}}:~|z|=1\}\), the space H is decomposed into the direct sum \(H=\chi _{_\eta }(U)H\oplus \chi _{_{\omega }}(U)H\) according to which U, N and |N| are decomposed as \(U= U_\eta \oplus U_\omega\), \(N=N_\eta \oplus N_\omega\) and \(|N|=|N_\eta |\oplus |N_\omega |\). Moreover, \(\sigma _{\mathbb {F}}(N_\eta )\) is included in the sector of the unit disc consisting of radii joining 0 to the points on \({\bar{\eta }}\). (Note that if \(\eta\) contains a portion of \(\sigma _{\mathbb {F}}(W)\backslash \sigma _{\mathbb {F}}(V)\), then its counterpart in \(\sigma _{\mathbb {F}}(N)\) is the singleton \(\{(0,0)\}\).)

4 Spectral measure

Once a Borel functional calculus is established, one can immediately construct a so-called Borel spectral measure satisfying (4.1)–(4.4) of the next theorem. We review the theory just to see how our improving Theorem 3.2 affects the spectral measure theorems.

Theorem 4.1

Let \(N=T+i\tau \in B(H)\) be a real selfadjoint operator or a complex normal operator, where \(T=(N+N^*)/2\), \(~i\tau =(N-N^*)/2~\) and H is a Hilbert space with the underlying field \({\mathbb {F}}\). (If \(\tau \ne 0\), then \({\mathbb {F}}\) is necessarily equal to \({\mathbb {C}}\).) Define \(E:{\mathcal {B}}\rightarrow B(H)\) by \(E(\gamma )=\chi _\gamma (N)\), where \({\mathfrak {B}}\) is the \(\sigma\)-algebra of all Borel subsets of \({\mathbb {F}}\). The following assertions are true for all \(\sigma ,\gamma ,\gamma _1,\gamma _2,\gamma _3,\ldots \in {\mathfrak {B}}\) whenever \(\gamma _m\cap \gamma _n=\emptyset\) for \(m\ne n\).

$$\begin{aligned}&E(\gamma )=E(\gamma )^2=E(\gamma )^*. \end{aligned}$$
(4.1)
$$\begin{aligned}&E({\mathbb {F}})=I,~E(\gamma )N=NE(\gamma ). \end{aligned}$$
(4.2)
$$\begin{aligned}&||E({\dot{\cup }}_{n=1}^\infty \gamma _n)x||^2=\sum _{n=1}^\infty ||E(\gamma _n) x||^2,~\forall x\in H.\end{aligned}$$
(4.3)
$$\begin{aligned}&\sigma _{\mathbb {F}}(N_\gamma )\subset {\bar{\gamma }}\cap \sigma _{\mathbb {F}}(N), \end{aligned}$$
(4.4)

where \(N_\gamma =N|_{E(\gamma )H}\).

Proof

Assertions (4.1)–(4.2) follow from Theorem 3.1 and the definition of E. Assertion (4.3) is an immediate consequence of \(\mu _x^N(\gamma )=||E(\gamma )x||^2.\) For (4.4), assume without loss of generality that \(E({\mathbb {F}}\backslash \gamma )H\ne \{0\}\) and observe that \(N=N_\gamma \oplus {\tilde{N}}_\gamma\) with respect to \(H=E(\gamma )H\oplus E({\mathbb {F}}\backslash \gamma )H.\) Then \(N_\gamma =N\chi _\gamma (N)|_{E(\gamma )H}\) and, hence, \(\sigma _{\mathbb {F}}(N_\gamma )\subset \sigma _{\mathbb {F}}(N\chi _\gamma (N))\). (Note that both \(N_\gamma\) and \(N\chi _\gamma (N)\) are normal.) Since \(N\chi _\gamma (N)=N_\gamma \oplus 0\), it follows from Remark 3.1 that

$$\begin{aligned} \sigma _{\mathbb {F}}(N_\gamma )\cup \{0\}= & {} \sigma _{\mathbb {F}}(N_\gamma \oplus 0)=\sigma _{\mathbb {F}}(N\chi _\gamma (N))= \overline{\cup _{x\in H}[\mu _x^N]\mathrm{ess.range}(\mathbf{id}\cdot \chi _\gamma )}\\= & {} \overline{\{\cup _x\mathrm{supp}(\mu _x^N|_\gamma )\}} \cup \{0\} \subset ({\bar{\gamma }}\cap \sigma _{\mathbb {F}}(N))\cup \{0\}. \end{aligned}$$

Assume \(0\in \sigma _{\mathbb {F}}(N_\gamma )\). Then, there exists a sequence of unit vectors \(x_n=E(\gamma )x_n\) such that \(N\chi _\gamma (N)x_n=N_\gamma x_n\rightarrow 0\). Hence, \(1=||x_n||^2=\int _\gamma d\mu _{x_n}^N\) and

$$\begin{aligned} \mathrm{dist}(0,\mathrm{supp}(\mu _{x_n}^N))^2\le \int _\gamma |s+it|^2d\mu _{x_n}^N=||Nx_n||^2\rightarrow 0. \end{aligned}$$

This implies that \(0\in \overline{\cup _{x\in H}\mathrm{supp}(\mu _x^N|_\gamma )}\subset {\bar{\gamma }}\cap \sigma _{\mathbb {F}}(N)\) and, thus, (4.4) is proven.\(\square\)

Definition 4.1

Let H be a real or complex Hilbert space and let \({\mathfrak {B}}\) denote the \(\sigma\)-algebra of all Borel subsets of the (underlying) field \({\mathbb {F}}\). Any triple \(({\mathbb {F}},{\mathfrak {B}}, E)\) is called a Borel spectral measure for a real selfadjoint operator or a complex normal operator \(N\in B(H)\) if \(E:{\mathfrak {B}}\rightarrow B(H)\) satisfies (4.1)–(4.4).

Theorem 4.2

If \(({\mathbb {F}},{\mathfrak {B}},E_j)\) is a Borel spectral measure for the real selfadjoint operator or the complex normal operator N \((j=1,2)\), then \(E_1=E_2\).

Proof

We can assume without loss of generality that \(E_1\) is equal to the spectral measure E of Theorem 4.1. Also, observe that, for \(j=1,2\),

$$\begin{aligned}&4\langle E_j(\gamma )x,y\rangle :=\nonumber \\&\langle E_j(\gamma )(x+y),x+y\rangle -\langle E_j(\gamma )(x-y),x-y\rangle \nonumber \\& + i\delta _{{\mathbb {F}},{\mathbb {C}}}[\langle E_j(\gamma )(x+iy),x+iy\rangle -\langle E_j(\gamma )(x-iy),x-iy\rangle ]. \end{aligned}$$
(4.5)

(Here, \(\delta _{\cdot ,\cdot }\) denotes the Kronecker \(\delta .)\) Letting \(M\in {\mathbb {N}}\) and choosing \(\gamma _n=\emptyset\) \(~(\forall n> M)\), it follows from (4.3) and (4.5) that

$$\begin{aligned} E_j(\emptyset )=0~\mathrm{~and~}~E_j(\cup _{n=1}^M\gamma _n) =\sum _{n=1}^ME_j(\gamma _n). \end{aligned}$$
(4.6)

In view of (4.1), every projection \(E_j(\gamma )\) is an orthogonal projection and, hence, it follows from [8] (page 18) that \(E_j(\gamma _h)E_j(\gamma _k)= E_j(\gamma _k) E_j(\gamma _h) =0\) whenever \(h\ne k\).

Now, define \(\nu _{j,x}(\gamma )=||E_j(\gamma )x||^2\). It follows from (4.3) and (4.6) that \(({\mathbb {F}},{\mathfrak {B}},\nu _{j,x})\) is a positive \(\sigma\)-additive measure \((j=1,2)\). Thus, to show \(E_1=E_2\), it is sufficient to show that \(\nu _{1,x}(\gamma )=\nu _{2,x}(\gamma )\) for all closed sets \(\gamma\). Note that, in view of (4.2) and Theorem 3.1, \(E_h(\gamma )E_k(\eta ) =E_k (\gamma )E_h (\eta )\) and \(E_h(\gamma )N=NE_h(\gamma )\) \(~\forall h,k=1,2\) and \(\forall \gamma ,\eta \in {\mathfrak {B}}\). Thus, for every \(\gamma \in {\mathfrak {B}}\),

$$\begin{aligned} H=H_{j,\gamma }\oplus H_{j,{\mathbb {F}}\backslash \gamma }, ~\mathrm{~where~}~ H_{j,\cdot }=E_j(\cdot )H. \end{aligned}$$

We claim \(H_{1,\gamma }=H_{2,\gamma }\) for every closed set \(\gamma\). Write \({\mathbb {F}}\backslash \gamma =\cup _{n=1}^\infty \eta _n\) for some monotone increasing sequence of open sets \(\eta _n\) satisfying \({\bar{\eta }}_n\cap \gamma =\emptyset\). Let \(x\in H_{1\gamma }\) and write \(y_n=E_2(\eta _n)x\). Then \(||E_2({\mathbb {F}}\backslash \gamma )x||^2=\lim _n||y_n||^2\). We prove \(y_n=0\). Fix n and let \({\mathcal {M}}:={\mathcal {Z}}(T,\tau ;y_n)\). It follows from the commutativity of \(N,E_1(\cdot )\) and \(E_2(\cdot )\) that \({\mathcal {M}}\subset E_2(\eta _n)H\cap E_1(\gamma )H\) and, hence, \(\sigma _{\mathbb {F}}(T|_{\mathcal {M}})\subset \gamma \cap {\bar{\eta }}_n=\emptyset\). Thus \({\mathcal {M}}=\{0\}\) and, therefore, \(E_1(\gamma )H\subset E_2(\gamma )H\). By a similar argument, \(E_2(\gamma )H\subset E_1(\gamma )H\) which implies that \(E_1(\gamma )=E_2(\gamma )\) for all closed sets \(\gamma\). Since \(E_j(\gamma )+E_j({\mathbb {F}}\backslash \gamma )=E_j({\mathbb {F}})=I\), it follows that \(E_1(\eta )=E_2(\eta )\) for all open sets \(\eta\). Thus, the two positive measures \(\nu _{1,x}\) and \(\nu _{2,x}\) are identical for all \(x\in H\). It follows from (4.5) that \(E_1\) and \(E_2\) are equal in the weak sense; i.e., \(\langle E_1(\gamma )x,y\rangle =\langle E_2(\gamma )x,y\rangle\) for all \(x,y\in H\).\(\square\)

Corollary 4.1

For the spectral measure \(({\mathbb {F}},{\mathfrak {B}}, E)\) of a real selfadjoint or complex normal operator N, the following assertions are true.

  1. 1.

    For every \(x\in H\), the set function \(\mu _x^N(\gamma )=||E(\gamma )x||^2\) is a positive \(\sigma\)-additive measure and

    $$\begin{aligned} ||f(T,\tau )x||^2=\int _{\sigma _{\mathbb {F}}(N)}|f(s,t)|^2d\mu _x^N(s,t),~\forall ~x\in H, ~\forall ~f\in L^\infty (N). \end{aligned}$$
  2. 2.

    For every \(x,y\in H\), the set function \(\mu _{x,y}^N(\gamma )=\langle E(\gamma )x,y\rangle\) is a complex \(\sigma\)-additive measure and

    $$\begin{aligned} \langle f(T,\tau )x,y\rangle =\int _{\sigma _{\mathbb {F}}(N)}f(s,t) d\mu _{x,y}^N(s,t),~\forall ~x,y\in H, ~\forall ~f\in L^\infty (N). \end{aligned}$$

5 Discrete normal operators

In the present section, we assume \(N\in B(H)\) is a real selfadjoint or a complex normal operator; we further assume the collection of its eigenvectors span H. In case H is separable, then it follows from the Gram–Schmidt process that there exists a finite or countable orthonormal basis \(\{e_n\}_{n\in {\mathbb {J}}}\) consisting of eigenvectors of N such that \(N=\sum _{n\in {\mathbb {J}}}\lambda _ne_n\otimes e_n\); more generally, if \(p\in {\mathbb {F}}[x]\), then \(p(N)=\sum _{n\in {\mathbb {J}}}p(\lambda _n)e_n\otimes e_n\). In fact, if \(f:{\mathbb {F}}\rightarrow {\mathbb {F}}\) is a bounded function, one can define \(f(N)=\sum _{n\in {\mathbb {J}}}f(\lambda _n)e_n\otimes e_n\in B(H)\). With no need of axiom of choice or Hausdorff maximality principle, we can partition \({\mathbb {J}}\) as \({\mathbb {J}}=\cup _{\lambda \in \varLambda }J_\lambda\) such that

$$\begin{aligned} H=\oplus _{\lambda \in \varLambda }{\mathcal {K}}(\lambda I-N), ~ f(N)=\oplus _{\lambda \in \varLambda } f(\lambda ) I_\lambda . \end{aligned}$$
(5.1)

In (5.1), one may either extend \(\varLambda\) to \({\mathbb {F}}\) by taking \({\mathcal {K}}(\lambda I-N)=\{0\}\) for the redundant \(\lambda\)’s or assume H is nonseparable and N has uncountably many eigenvalues. In either case, there is a need for the definition of uncountable direct sums and extended Bessel inequalities or extended Parseval equalities. The following definition is surprisingly simple. Recall that an orthogonal projection \(P\in B(H)\) is characterized as \(P=P^*=P^2\); such an orthogonal projection decomposes the space as the orthogonal direct sum \(H=PH\oplus (I-P)H\). To avoid any kind of confusion between orthogonality term used for matrices or projection, we use the term perpendicularity for linear subspaces. Also, recall that the notation \(\oplus\) in this paper is solely used for orthogonal direct sum; i.e., the expression \(\oplus _{\alpha \in \varLambda }M_\alpha\) necessarily implies that \(M_\alpha\) and \(M_\beta\) are mutually perpendicular (for all \(\alpha \ne \beta\) in \(\varLambda\)).

However, we acknowlege that until Theorem 5.2, the results are known for which we refer to Shirbisheh [13].

Definition 5.1

A subspace M of H is said to be the orthogonal direct sum of a family \(\{M_\alpha \}_{\alpha \in \varLambda }\) of mutually perpendicular subspaces \(M_\alpha\), and write \(M=\oplus _{\alpha \in \varLambda }M_\alpha\), if M is the smallest closed subspace containing \(\cup _{\alpha \in \varLambda }M_\alpha\).

To justify our definition of orthogonal direct sum, we prove the following theorem which extends Bessel’s inequality and Parseval’s equality for families of mutually perpendicular closed subspaces. We could not persuade ourselves to skip the proof; instead, we avail ourselves of acknowledging that it is somehow known to the experts. Also, we have to mention that the terms extended Bessel’s inequality and extended Parseval’s equality are new.

Theorem 5.1

Let \(\{M_\alpha :~\alpha \in \varLambda \}\) be a family of mutually perpendicular subspaces of a Hilbert space H over the field \({\mathbb {F}}\) and let \(P_\alpha \in B(H)\) be the orthogonal projection onto \(M_\alpha\). Let \(M=\oplus _{\alpha \in \varLambda } M_\alpha\) with the corresponding orthogonal projection \(P\in B(H)\). Then the following extended Parseval’s equality and extended Bessel’s inequality hold:

$$\begin{aligned} \sup \{\sum _{\alpha \in L}||P_\alpha x||^2:~ L~\mathrm{finite}~\subset \varLambda \}=||Px||^2\le ||x||^2~ \forall ~x \in H. \end{aligned}$$

Also, the set \(\varLambda _x:=\{\alpha \in \varLambda :~P_\alpha x\ne 0\}\) has an identification with \({\mathbb {N}}\) or some finite initial segment \(\{1,2,\ldots ,n\}\), independent of which the following conditions (a)–(c) hold.

  1. (a)

    The Bessel inequality and the Parseval equality can be modified as follows:

    $$\begin{aligned} \sum _{\alpha \in \varLambda _x}||P_\alpha x||^2= ||Px||^2\le ||x||^2~{~}~(\forall x\in H). \end{aligned}$$
  2. (b)

    The series \(Px=\sum _{\alpha \in \varLambda _x} P_\alpha x~\) is convergent in norm \((\forall x\in H)\).

  3. (c)

    The numerical series \(\langle Px,y\rangle =\sum _{\alpha \in \varLambda _x}\langle P_\alpha x,y\rangle\) is convergent \((\forall x,y\in H)\).

Proof

Fix \(x\in H\). For each \(n\in {\mathbb {N}}\), let \(\varLambda _{x,n}=\{\alpha \in \varLambda :~||P_\alpha x||\ge 1/n\}\). Then, for any finite subset \(\{\alpha _1,\alpha _2,\dots ,\alpha _m\}\) of \(\varLambda _{x,n}\),

$$\begin{aligned} m/n^2 \le \sum _{k=1}^m||P_{\alpha _k}x||^2= \sum _{k=1}^m\langle P_{\alpha _k}x,x\rangle =\langle (P_{\alpha _1}+\cdots +P_{\alpha _m})x,x\rangle \le ||x||^2. \end{aligned}$$
(5.2)

Thus, \(m\le n^2||x||^2\), which shows that \(\varLambda _{x,n}\) is finite and, hence, \(\varLambda _x=\cup _{n\in {\mathbb {N}}}\varLambda _{x,n}\) is countable. For the sake of uniformity, we assume without loss of generality that \(\{M_\alpha \}_{\alpha \in \varLambda }\) contains infinitely many copies of \(\{0\}\); this enables us to identify each \(\varLambda _x\) with \({\mathbb {N}}\) by supplementing it with enough copies of \(\{0\}\), if necessary. Now, let L be any finite subset of \(\varLambda\). Using the argument leading to (5.2) yields \(\sum _{\alpha \in L}||P_\alpha x||^2\le ||x||^2\), which establishes the extended Bessel’s inequality as well as its discrete form in (a). Note that the positive series in (a) is absolutely convergent and, hence, does not depend on the order of \(\varLambda _x\).

To prove extended Parseval equality, observe that

$$\begin{aligned} ||\sum _{k=m+1}^nP_kx||^2= & {} \langle \sum _{k=m+1}^nP_kx,x\rangle =\sum _{k=m+1}^n \langle P_kx,x\rangle \\= & {} \sum _{k=m+1}^n||P_k x||^2\rightarrow 0~\mathrm{~ as~}~m,n\rightarrow \infty . \end{aligned}$$

Therefore, there exists \(\xi \in M\) such that the series \(\sum _{n=1}^\infty P_nx\) and \(\sum _{n=1}^\infty ||P_nx||^2\) converge to \(\xi\) and \(||\xi ||^2\), respectively. Also, the arguments leading to (5.2) reveal that Parseval equality and conclusions (b)-(c) hold if x is replaced by \(\xi =P\xi\). To complete the proof, we claim \(\xi =Px\) or, equivalently, \(x-\xi \perp M\). This, in turn, is equivalent with showing that \(x-\xi \perp M_\beta\) for all \(\beta \in \varLambda _x\). Fix \(\beta \in \varLambda\) and choose arbitrary \(y=P_\beta y\in M_\beta\). Assume without loss of generality that \(\beta \in \varLambda _x\) and observe that

$$\begin{aligned} \langle x-\xi ,P_\beta y\rangle= & {} \langle P_\beta (x-\xi ),y\rangle =\langle P_\beta x,y\rangle - \langle P_\beta \sum _{n=1}^\infty P_nx,y\rangle \\= & {} \langle P_\beta x,y\rangle -\langle \sum _{n=1}^\infty P_\beta P_nx,y\rangle =\langle P_\beta x,y\rangle -\langle P_\beta x,y\rangle =0. \end{aligned}$$

Thus, \(x-\xi \perp M_\beta\) and, by linearity and continuity of the inner product in the first component, \(x-\xi \perp M\). \(\square\)

The following corollary provides a list of what we expect from an orthogonal direct sum.

Corollary 5.1

With the notation of the theorem, let \(x_\alpha \in M_\alpha\) for all \(\alpha \in \varLambda\) and let \(x\in \oplus _{\alpha \in \varLambda }M_\alpha\). The following assertions are equivalent.

  1. (i)

    \(x_\alpha =P_\alpha x~\) for all \(\alpha \in \varLambda\).

  2. (ii)

    \(x_\alpha =P_\alpha x~\) for all \(\alpha \in \varLambda\) and \(\sup \{\sum _{\alpha \in F}||x_\alpha ||^2:~F~\mathrm{finite ~subset~ of}~\varLambda \}=||x||^2\).

  3. (iii)

    The set \(\varLambda _x:=\{\alpha \in \varLambda :~x_\alpha \ne 0\}~\) is countable and \(x=\sum _{\alpha \in \varLambda _x}x_\alpha\).

  4. (iv)

    The vector x is uniquely represented as \(x=\sum _{\alpha \in \varLambda }x_\alpha ~\) in the sense that for all \(\epsilon >0\), there exists a finite subset F of \(\varLambda\) such that \(||x-\sum _{\alpha \in L}x_\alpha ||<\epsilon\) whenever \(F\subset L\subset \varLambda\) and \(\mathrm{card}(L)<\infty\).

The next corollary is an extension of the Pythagorean Theorem.

Corollary 5.2

(Extended Pythagorean Theorem) Let \(\{M_\alpha \}_{\alpha \in \varLambda }\) be a family of closed linear subspaces of the Hilbert space H with the underlying field \({\mathbb {F}}\). Let \(P_\alpha\) denote the orthogonal projection onto \(M_\alpha\) \((\alpha \in \varLambda )\). Then, the following assertions are equivalent.

  1. (a)

    \(M_\alpha \perp M_\beta\) for all distinct \(\alpha ,\beta \in \varLambda\).

  2. (b)

    \(P_\alpha P_\beta =0~\) for all distinct \(\alpha ,\beta \in \varLambda\).

  3. (c)

    \(P_\alpha +P_\beta\) is an orthogonal projection for all distinct \(\alpha ,\beta \in \varLambda\).

  4. (d)

    \(||x_\alpha +x_\beta ||^2=||x_\alpha ||^2+||x_\beta ||^2\) whenever \(x_\alpha \in M_\alpha\), \(x_\beta \in M_\beta\) and \(\alpha \ne \beta\).

Proof

The equivalence of (a) and (b) follows from the simple verification of \(\langle P_\alpha P_\beta x,\) \(y\rangle\) for all \(x,y\in H\). The equivalence of (b) and (c) is verified on page 18 of [8]. The proof of (a) \(\Rightarrow\) (d) is a straightforward computation. For the converse, we must show that \(\langle x_\alpha ,x_\beta \rangle =0\) whenever \(\alpha \ne \beta \in \varLambda\), \(x_\alpha \in M_{\alpha }\) and \(x_\beta \in M_\beta\). Assume \(\langle x_\alpha ,x_\beta \rangle \ne 0\) to reach a contradiction. Let \(z=\langle x_\beta ,x_\alpha \rangle /|\langle x_\alpha ,x_\beta \rangle |\). Then

$$\begin{aligned} ||zx_\alpha ||^2+||x_\beta ||^2=||zx_\alpha +x_\beta ||^2=||zx_\alpha ||^2+||x_\beta ||^2+2~\mathfrak {R}e(z\langle x_\alpha ,x_\beta \rangle ) \end{aligned}$$

and, hence, \(0=2\mathfrak {R}e |\langle x_\alpha ,x_\beta \rangle |=2 |\langle x_\alpha ,x_\beta \rangle |\ne 0\); a contradiction.\(\square\)

We are now ready to state our main theorem about the functional calculus of discrete normal operators. For the definitions of Borel functional calculus \(f\mapsto f(N)\) and Borel spectral measure E we refer to Theorems 3.1 and 4.1. Also, we fix the notation \(I_\lambda\) for the identity operator on the subspace \({\mathcal {K}}(\lambda I-N)\).

Theorem 5.2

(Discrete spectral measure) Let \(N\in B(H)\) be a real or complex normal operator whose domain H is the smallest closed subspace containing the collection of all eigenvectors of N corresponding to the eigenvalues \(\sigma _p(N)\subset \sigma _{\mathbb {F}}(N)\). Let \(B(\sigma _p(N))\) denote the collection of all functions \(f:{\mathbb {F}}\rightarrow {\mathbb {F}}\) such that \(f|_{\sigma _p(N)}\) is bounded. Then \(L^\infty (N)\subset B(\sigma _p(N))\) and

$$\begin{aligned} ||f||_N=||f||_{p,N},~\mathrm{~ where~}~||f||_{p,N}:=\sup \{|f(\lambda )|:~\lambda \in \sigma _p(N)\}. \end{aligned}$$
(5.3)

Moreover, the functional calculus \(f\mapsto f(N):B(\sigma _p(N)\rightarrow B(H)\) and the spectral measure \(({\mathbb {F}},2^{\mathbb {F}},{\tilde{E}})\) are, respectively, isometric extensions of the Borel functional calculus \(f\mapsto f(N):L^\infty (N)\rightarrow B(H)\) and the Borel spectral measure \(({\mathbb {F}},{\mathfrak {B}},E)\), where

$$\begin{aligned} f(N)= & {} \oplus _{\lambda \in \sigma _p(N)}f(\lambda )I_\lambda ,~{~}~\forall ~f\in B(\sigma _p(N)), \end{aligned}$$
(5.4)
$$\begin{aligned} {\tilde{E}}(\gamma )= & {} \oplus _{\lambda \in \{\sigma _p(N)\cap \gamma \}}I_\lambda ,~{~}~\forall ~\gamma \in 2^{\mathbb {F}}. \end{aligned}$$
(5.5)

Proof

Choose arbitrary \(f\in L^\infty (N)\) and \(\epsilon >0\). There exists \(x\in H\) such that \(||f||_N<||f||_{L^\infty (\mu _x^N)}+\epsilon\). Let \(x_\lambda\) be the projection of x into \({\mathcal {K}}(\lambda I-T)\). Then, for all \(\gamma \in {\mathfrak {B}}\),

$$\begin{aligned} \mu _x^N(\gamma )=||\chi _\gamma (N)x||^2=\sum _{\lambda \in \sigma _p(N)}||\chi _\gamma (N)x_\lambda ||^2= \sum _{\lambda \in \gamma }||x_\lambda ||^2=\sum _{\lambda \in \gamma }\mu _x^N(\{\lambda \}). \end{aligned}$$
(5.6)

Choose \(\lambda \in \sigma _p(N)\) such that \(||f||_{L^\infty (\mu _x^N)}<|f(\lambda )|+\epsilon\). It follows that \(||f||_N<|f(\lambda )|+2\epsilon <||f||_{\sigma _p(N)}+2\epsilon\). Letting \(\epsilon \rightarrow 0\) yields \(||f||_N\le ||f||_{p,N}\).

For the converse, choose arbitrary \(\lambda \in \sigma _p(N)\) and let \(Nx=\lambda x\) for some unit vector \(x\in H\). Then \(\mu _x^N(\{\lambda \})\ne 0\) and, hence, \(|f(\lambda )|\le ||f||_{L^\infty (\mu _x^N)}\le ||f||_N\). Thus, \(||f||_{p,N}\le ||f||_N\) which proves \(L^\infty (N)\subset B(\sigma _p(N))\) and completes the proof of (5.3).

To prove (5.4), let \(f\in L^\infty (N)\) and apply (5.6) to conclude that

$$\begin{aligned} ||f(N)x||^2= & {} \int |f|^2d\mu _x^N=\sum _{\lambda \in \varLambda _x}\int _{\{\lambda \}}|f(\lambda )|^2d\mu _x^N\\= & {} \sum _{\lambda \in \varLambda _x}||f(\lambda ) I_\lambda x||=\sum _{\lambda \in \sigma _p(N)}||f(\lambda ) I_\lambda x||. \end{aligned}$$

Now, in view of (4.5), the assertion (5.4) follows. It is now easy to see that the functional calculus (5.4) is a well-defined isometric \(*\)-algebra isomorphism which extends the Borel functional calculus.

For (5.5), it follows from (5.4) that

$$\begin{aligned} E(\gamma )= & {} \chi _\gamma (N)=\oplus _{\lambda \in \sigma _p(N)}\chi _\gamma (\lambda )I_\lambda \\= & {} \oplus _{\lambda \in \{\sigma _p(N)\cap \gamma \}}\chi _\gamma (\lambda )I_\lambda ={\tilde{E}}(\gamma ) \end{aligned}$$

Therefore, \({\tilde{E}}\) is an extension of E. It is easy to see that \(({\mathbb {F}},2^{\mathbb {F}},{\tilde{E}})\) is a spectral measure.\(\square\)