n-Harmonic calculus and applications

Abstract

We define and study a functional calculus for vector valued n-harmonic functions. As applications, we obtain a generalization of the well-known von-Neumann’s inequality to several variables. We also use weighted algebras analogues of the classical theorems of N. Wiener and P. Lévy on absolutely Fourier series in order to obtain multi-dimensional versions of N. Wiener and P. Lévy theorems.

1 Preliminaries and introduction

Let $(A,\left\| .\right\| )$ be a complex algebra with unit. If $x\in A$ the symbols $Sp_{A}(x)$ and $\rho _{A}(x)$ denote the spectrum of x and its spectral radius, respectively. Let $x\longmapsto x^{*}$ be an involution on A.An element h of A is called hermitian if $h^{*}=h$. The set of all hermitian elements of A will be denoted by H(A). The real and imaginary parts of an element x of A are denoted by Rex and Imx, respectively, i.e., $\textit{Re}x=\left( x+x^{*}\right) /2$, $Im x=\left( x-x^{*}\right) /2i.$ We say that a Banach algebra $A\$is hermitian if the spectrum of every element of H(A) is real ([12], Definition 5.1, p. 23). For elements $h\$and k of H(A),  we write $h\ge k$ to indicate that $h-k$ is positive, i.e., $Sp_{A}(h-k)\subset [ 0,+\infty [$. Let x be an element of A. We denote by $\left| x\right|$ the square root of the spectral radius of the element $x^{*}x$, i.e., $\left| x\right| =\rho _{A}(x^{*}x)^{ \frac{1}{2}}$. In ([12], Theorem 5.2, (5.4) and (5.8), p. 23-25), V. Pt àk proved the following result: If A is hermitian, then $\left| .\right|$ is an algebra seminorm on A such that $\rho _{A}(x)\le \left| x\right|$, for every $x\in A.$ The following result of Shirali- Ford ([16], Theorem 1, p. 275) will be needed throughout the paper:

$$\begin{aligned} A\ \text {is hermitian}\Longrightarrow x^{*}x\ge 0\text {, for every } x\in A. \end{aligned}$$

(1)

Throughout the paper, e will denote the unit of A, and for scalars r we often write simply r for the element re of A. Also Sp(A) denotes the spectrum of A, that is the set of non-zero continuous characters of A. Let n be a positive integer and let $A^{n}$ denote the cartesian product of n copies of A. Let $\textbf{a}=\left( a_{1},\ldots ,a_{n}\right) \in A^{n}$ be a commutative family of elements of A (a c.f.e. in short). Then the full sub-algebra B generated by $\textbf{a}$ is a unital commutative algebra. Let $\widehat{\textbf{a}}$ denote the Gel’fand transformation defined by:

$$\begin{aligned} \widehat{\textbf{a}}\left( \chi \right) =\left( \chi \left( a_{1}\right) ,\ldots ,\chi \left( a_{n}\right) \right) \in \mathbb {C}^{n}\text {, for every } \chi \in Sp\left( B\right) . \end{aligned}$$

The image $\widehat{\textbf{a}}\left( SpB\right) \subset \mathbb {C}^{n}$ is therefore a nonempty compact subset of $\mathbb {C}^{n}$. It is called the simultaneous spectrum or the joint spectrum of $\textbf{a}$ and denoted by $Sp\left( A,\textbf{a}\right)$ or just $Sp\left( \textbf{a} \right)$ ([2], Definition 7, p. 100). One has $Sp\left( \textbf{a}\right) \subset \prod \limits _{i=1}^{n}Sp_{A}\left( a_{i}\right)$.

Let A be a complex unital Banach algebra and $A^{\prime }\$the topological dual of A. Let $\Omega$ be an open subset of $\mathbb {C}^{n}$ and $f:\Omega \longrightarrow A$ be an A-valued function. Then f is said to be holomorphic if $\varphi (f(z))$ is holomorphic on $\Omega$ in the classical sense for every $\varphi \in A^{\prime }$. The set of all holomorphic$\ A$-valued functions on $\Omega \$isdenoted by $\mathcal {H} \left( \Omega ,A\right)$.Since the dual $A^{\prime }$ separates the points of A, the most results of complex function theory ([13, 14]), such as Cauchy’s integral, Taylor expansion, Cauchy estimates and so on, are applicable to $\mathcal {H}\left( \Omega ,A\right)$. It is clear that $\mathcal {H}\left( \Omega ,A\right)$ is a complex unital algebra. Moreover, if f is an element of $\mathcal {H}\left( \Omega ,A\right)$ and if $f(z)\$ is invertible for every $z\in \Omega$, then the function $f^{-1}$ defined by $f^{-1}\left( z\right) =f(z)^{-1}$ for each $z\in \Omega$ is an element of $\mathcal {H}\left( \Omega ,A\right) .$

A continuous A-valued function $f:\Omega \longrightarrow A$ is a said to be n-harmonic if f is harmonic in each complex variable that is if $z_{j}=x_{j}+iy_{j}$, f should satisfy the n differential equations:

$$\begin{aligned} \frac{\partial ^{2}f}{\partial ^{2}x_{j}}+\frac{\partial ^{2}f}{\partial ^{2}y_{j}}=0,\ \text { for }j=1,\ldots ,n. \end{aligned}$$

The set of all n-harmonic A-valued functions on $\Omega$ is denoted by $h\left( \Omega ,A\right)$. If f is holomorphic on $\Omega$, then it is holomorphic in each variable, so we have $\mathcal {H}\left( \Omega ,A\right) \subset h\left( \Omega ,A\right)$. Let A be an involutive complex Banach algebra. An A-valued function $f:\Omega \longrightarrow A$ is said to be hermitian if $f(z)=f(z)^{*}$, for every $z\in \Omega$. We denote by Ref (resp. Imf) the real part of f (resp. the imaginary part of f) defined by $Re f(z)=Re\left( f(z)\right)$ (resp. $Imf(z)=Im\left( f(z)\right)$), for every $z\in \Omega$.

Let $z_{0}\in \mathbb {C}$ and $r>0$, the open (resp. closed) disc with center $z_{0}$ and radius r is denoted by $D\left( z_{0},r\right)$ (resp. $\overline{D}\left( z_{0},r\right)$); its boundary, denoted by $T\left( z_{0},r\right)$, is the circle with center $z_{0}$ and radius r. If $z^{0}=\left( z_{1}^{0},\ldots ,z_{n}^{0}\right) \in \mathbb {C}^{n}$ and $r=\left( r_{1},\ldots ,r_{n}\right)$ is a multi-index $\left( \mathbb {R} _{+}^{*}\right) ^{n}$, the open polydisc with center $z^{0}$ and radius r is the set

$$\begin{aligned} D^{n}\left( z^{0},r\right) =\prod \limits _{j=1}^{n}D\left( z_{j}^{0},r_{j}\right) . \end{aligned}$$

Itsclosureis denoted by $\overline{D}^{n}\left( z^{0},r\right)$. The notation $T^{n}\left( z^{0},r\right) \$denotes the torus of $\mathbb { C}^{n}\$with center $z^{0}$ and radius r, that is $T^{n}\left( z^{0},r\right) =\prod \nolimits _{j=1}^{n}T\left( z_{j}^{0},r_{j}\right)$. Let $\mathbb {Z}^{+}=\left\{ 0,1,2,..\right\}$.A multi-index $\alpha$ is an element of $\left( \mathbb {Z}^{+}\right) ^{n}$. If$\ \alpha =\left( \alpha _{1},\ldots ,\alpha _{n}\right)$ is a multi-index and $w=\left( w_{1},\ldots ,w_{n}\right) \in \mathbb {C}^{n}$, we write $w^{\alpha }$ for the monomial (power product) $w_{1}^{\alpha _{1}}\ldots w_{n}^{\alpha _{n}}$. We also consider the following differential form, on $\mathbb {C}^{n}$, $dz=dz_{1}\ldots dz_{n}$. In the sequel, all algebras considered here are complex and unital ones.

The A-valued harmonic functional calculus for an element of an involutive Banach algebra is defined and studied in [5]. Here we construct an A -valued n-harmonic calculus for an arbitrary n-tuple of elements of an involutive Banach algebra elements. This calculus consists in giving a sense to $f(\mathbf {a)}$ whenever $\mathbf {a=}\left( a_{1},\ldots ,a_{n}\right) \in A^{n}$ and f is an A-valued n-harmonic function on a neighbourhood U of the simultaneous spectrum $Sp(\textbf{a})$ of $\textbf{a}$. To that aim, we need to introduce a functional calculus for holomorphic A-functions. The paper is organized as follows. In Sect. 2, we introduce a vector-valued Cauchy transform. This allows us to introduce a functional calculus for holomorphic A-functions. We show that this calculus is continuous (namely the mapping $f\longmapsto C\left[ f\right] \left( \textbf{ a}\right) \$is continuous) and satisfies the spectral mapping theorem . Sect. 3 relies highly on Sect. 2 where we define and study a vector-valued n-harmonic calculus. The most important properties of this calculus are studied. The last section is devoted to applications. The first one is a generalization of the well-known von-Neumann’s inequality to several variables. The second concerns the classical and famous theorems of N. Wiener [19] and P. Lévy [11].

For more details on holomorphic functional calculus, we refer the reader to [1, 18].

2 A vector-valued Cauchy transform

We first introduce a vector-valued Cauchy transform by means of appropriate vector-valued kernel. In particular, we obtain a functional calculus for holomorphic A-functions which will be useful to us later.

Definition 2.1

Let A be a complex unital Banach algebra, $\Omega$ an open subset of $\mathbb {C}^{n}$, $z^{0}=\left( z_{1}^{0},\ldots ,z_{n}^{0}\right) \in \Omega$, $r=\left( r_{1},\ldots ,r_{n}\right) \in \left( \mathbb {R}_{+}^{*}\right) ^{n}$ such that $\overline{D}^{n}\left( z^{0},r\right) \subset \Omega$. If $f\in \mathcal {H}\left( \Omega ,A\right)$ and $\textbf{a}=\left( a_{1},\ldots ,a_{n}\right) \in A^{n}$ be a c.f.e. with $Sp(\mathbf {a)\subset } D^{n}\left( z^{0},r\right)$, then

$$\begin{aligned} f(\textbf{a})=\frac{1}{\left( 2\pi i\right) ^{n}}\int _{T^{n}\left( z^{0},r\right) }f(z)C(\textbf{a},z)dz, \end{aligned}$$

where

$$\begin{aligned} C(\textbf{a},z)=\prod \limits _{j=1}^{n}(z_{j}-a_{j})^{-1}=\prod \limits _{j=1}^{n}C(a_{j},z_{j}). \end{aligned}$$

If we denote by $\Phi _{\textbf{a}}(f)$ the element $f(\textbf{a})$, one has a linear mapping of $\mathcal {H}\left( \Omega ,A\right)$ into A, denoted by:

$$\begin{aligned} \Phi _{\textbf{a}}:\mathcal {H}\left( \Omega ,A\right) \longrightarrow A:f\longmapsto f(\textbf{a}) \end{aligned}$$

(2)

Suppose f is a function on $\Omega$ into A and $\textbf{a}=\left( a_{1},\ldots ,a_{n}\right) \in A^{n}$. The function $\textbf{a}f$ is defined by:

$$\begin{aligned} \textbf{a}f(z)=\left( a_{1}f(z),\ldots ,a_{n}f(z)\right) ,\text { for every }z\in \Omega . \end{aligned}$$

We say that $\textbf{a}$ and f are commuting if $\textbf{a}f(z)=f(z) \textbf{a}$ for all z in $\Omega$ that is:

$$\begin{aligned} \left( a_{1}f(z),\ldots ,a_{n}f(z)\right) =\left( f(z)a_{1},\ldots ,f(z)a_{n}\right) ,\text { for every }z\in \Omega . \end{aligned}$$

If $\textbf{a}$ commutes with every element of $\mathcal {H}\left( \Omega ,A\right)$, then we say that $\textbf{a}$ and $\mathcal {H}\left( \Omega ,A\right)$ commute. Now $f,g\in \mathcal {H}\left( \Omega ,A\right)$ and $\textbf{a}=\left( a_{1},\ldots ,a_{n}\right)$ be a c.f.e. as described in Definition 2.1. Then as in the classical case, one has $\left( fg\right) ( \textbf{a})=f(\textbf{a})g(\textbf{a})$ if $\textbf{a}$ and g are commuting. Unless otherwise stated we assume that $\textbf{a}$ and $\mathcal { H}\left( \Omega ,A\right)$ are commuting.

Let $\mathcal {C(}T^{n},A)$, where $T^{n}=T^{n}\left( 0,1\right)$, be the algebra of all A-valued continuous functions on $T^{n}$.For every $f\in \mathcal {C}\left( T^{n},A\right)$, we define

$$\begin{aligned} C\left[ f\right] \left( \textbf{a}\right) =\frac{1}{\left( 2\pi i\right) ^{n} }\int _{T^{n}\left( z^{0},r\right) }f(z)C(\textbf{a},z)dz. \end{aligned}$$

The mapping $\mathcal {C}\left( T^{n},A\right) \longrightarrow A:f\longmapsto C\left[ f\right] \left( \textbf{a}\right)$ is obviously a linear map. Now, since the mapping $z\longmapsto \left\| C(\textbf{a},z)\right\|$ is continuous, and therefore bounded, on $T^{n}\left( z^{0},r\right)$, there exists a positive constant M such that:

$$\begin{aligned} \left\| C\left[ f\right] \left( \textbf{a}\right) \right\| \le M\left| f\right| _{T^{n}\left( z^{0},r\right) }\text {,}\ \text {for every }f\in \mathcal {C}\left( T^{n}(z^{0},r),A\right) , \end{aligned}$$

where

$$\begin{aligned} \left| f\right| _{T^{n}\left( z^{0},r\right) }=\sup \left\{ \left\| f(z)\right\| :z\in T^{n}(z^{0},r)\right\} . \end{aligned}$$

It follows that the mapping

$$\begin{aligned} f\longmapsto C\left[ f\right] \left( \textbf{a}\right) \end{aligned}$$

is continuous from $\left( \mathcal {C}\left( T^{n},A\right) ,\text { } \left| .\right| _{T^{n}\left( z^{0},r\right) }\right)$ into $\left( A,\left\| .\right\| \right)$. It is called the Cauchy transform of f at $\textbf{a}$ ([17], p. 49). If $\textbf{a}=\left( a_{1},\ldots ,a_{n}\right) \in A^{n}$ be a c.f.e. with $Sp(\mathbf {a)\subset } D^{n}\left( 0,1\right)$, then for every $f\in \mathcal {H}(\Omega ,A)$, we have $C\left[ f\right] \left( \textbf{a}\right) =\Phi _{\textbf{a}}(f)$. Moreover if $P(z)=z_{1}^{\gamma _{1}}\ldots z_{n}^{\gamma _{n}}$ is a monomial, then $C\left[ P\right] \left( \textbf{a}\right)$ is actually $a_{1}^{\gamma _{1}}\ldots a_{n}^{\gamma _{n}}$, where $\gamma _{1},..,\gamma _{n}\in \mathbb {Z} _{+}$. Whence for every analytic polynomial P, one has $C\left[ P\right] \left( \textbf{a}\right) =P(\textbf{a}).\$So we have the following:

Theorem 2.2

Let A be a complex unital Banach algebra, $\Omega$ an open subset of $\mathbb {C}^{n}$, $z^{0}=\left( z_{1}^{0},\ldots ,z_{n}^{0}\right) \in \Omega$, $r=\left( r_{1},\ldots ,r_{n}\right) \in \mathbb {R}_{+}^{n}$ such that $\overline{ D}^{n}\left( z^{0},r\right) \subset \Omega$ and $\textbf{a}=\left( a_{1},\ldots ,a_{n}\right) \in A^{n}$ be a c.f.e. with

$$\begin{aligned} Sp\mathbf {(\mathbf {a)}\subset }D^{n}\left( z^{0},r\right) . \end{aligned}$$

Then there exists a continuous linear map $\Theta _{\textbf{a}}$ from $\mathcal {C}\left( T^{n}(z^{0},r),A\right)$ into A with the following properties:

(1) For every analytic polynomial P, one has

$$\begin{aligned} \Theta _{\textbf{a}}\left( P\right) =P(\textbf{a}) \end{aligned}$$

(2) $\Theta _{\textbf{a}}/_{\mathcal {H}\left( \Omega ,A\right) }$ is multiplicative and $\Theta _{\textbf{a}}\left( z_{j}\right) =a_{j}$, $(j=1,\ldots ,n)$, here $z_{j}$ denotes the j-th coordinate projection $\mathbb { C}^{n}\longrightarrow \mathbb {C}$.

Remark 2.3

(1) Let $\Omega =D^{n}\left( 0,R\right) ,$ where $R=\left( R_{1},\ldots ,R_{n}\right) \in \mathbb {R}_{+}^{n}$, and $f\in \mathcal {H}\left( \Omega ,A\right) \$with the the Taylor expansion

$$\begin{aligned} f(z)=\sum _{\alpha }a_{\alpha }z^{\alpha }\text {, for every }z\in \Omega , \end{aligned}$$

where $\left( a_{\alpha }\right)$ is a sequence in A. If $x=\left( x_{1},\ldots ,x_{n}\right) \in A^{n}$ be c.f.e. with $\rho \left( x_{j}\right) <R_{j}\ (j=1,\ldots ,n)$, then

$$\begin{aligned} f(\textbf{a})=\sum _{\alpha }a_{\alpha }x^{\alpha }. \end{aligned}$$

(2) Let $f\in \mathcal {H}(\Omega ,A)$ be such that $f(z)^{-1}$ exists, for every $z\in \Omega$. Then $f^{-1}(x)=f(x)^{-1}$, i.e.,

$$\begin{aligned} f(x)^{-1}=\frac{1}{\left( 2\pi i\right) ^{n}}\int _{T^{n}\left( z^{0},r\right) }f(z)^{-1}C(x,z)dz. \end{aligned}$$

Let $\Omega _{1}$ and $\Omega _{2}\$be two open subsets of $\mathbb {C}$. Suppose $f\in \mathcal {H}\left( \Omega _{1},A\right)$ and $g\in \mathcal {H} \left( \Omega _{2},A\right)$ satisfy the condition that for every compact set $K_{2}$ in $\Omega _{2}$, there exists a compact $K_{1}$ in $\Omega _{1}$ such that $Sp_{A}\left( g(z)\right)$ is contained in $K_{1}$, for every z in $K_{2}$. Let $z_{0}\in \Omega _{2}$ and $r\in \mathbb {Z}_{+}$ such that $\overline{D}\left( z_{0},r\right) \subset \Omega _{2}$. Let $K_{1}$ be a compact set in $\Omega _{1}$, which contains each $Sp_{A}\left( g(z)\right) \$for every $z\in \overline{D}\left( z_{0},r\right)$. As in ([2], Proposition 3, p. 29), we choose a suitable positively oriented simple closed rectifiable contour $\Gamma _{1}$ such that the interior domain $int(\Gamma _{1})\$of $\Gamma _{1}$ contains $K_{1}$ and $int(\Gamma _{1})\cup \Gamma _{1}\subset \Omega _{1}$. Then for $z\in \overline{D} \left( z_{0},r\right)$, we have

$$\begin{aligned} Sp_{A}(g(z))\subset K_{1}\subset int(\Gamma _{1})\cup \Gamma _{1}\subset \Omega _{1}. \end{aligned}$$

It follows, from ([2], Proposition 4, p. 29), that:

$$\begin{aligned} f\left( g(z)\right) =\frac{1}{2\pi i}\int _{\Gamma _{1}}f(w)\left( we-g(z)\right) ^{-1}dw \end{aligned}$$

(3)

Since for any fixed w on $\Gamma _{1}$,

$$\begin{aligned} z\longmapsto \left( we-g(z)\right) ^{-1} \end{aligned}$$

is analytic on $D\left( z_{0},r\right)$, it follows from 2) of Remark 2.3 that, for $w\in \Gamma _{1}$ and $z\in D\left( z_{0},r\right)$,

$$\begin{aligned} \left( we-g(z)\right) ^{-1}=\frac{1}{2\pi i}\int _{T\left( z_{0},r\right) }\left( we-g(u)\right) ^{-1}(u-z)^{-1}du \end{aligned}$$

which shows that, for every $z\in D\left( z_{0},r\right)$, one has

$$\begin{aligned} f\left( g(z)\right) =\left( \frac{1}{2\pi i}\right) ^{2}\int _{\Gamma _{1}}\int _{T\left( z_{0},r\right) }f(w)\left( we-g(u)\right) ^{-1}(u-z)^{-1}dudw. \end{aligned}$$

Let $\varphi$ be any bounded functional on A. As in ([20], Lemma 2.4, p. 297), we can prove that $\varphi \left( f\left( g(z)\right) \right)$ is analytic on $D\left( z_{0},r\right)$. It follows, from ([15], Definition 3. 30, p. 78), that the "composite function" $h=f\circ g$ defined by $h(z)=f(g(z))\$for every $z\in$ $\Omega _{2}$ is an element of $\mathcal {H} \left( \Omega _{2},A\right)$.More generally, we obtain a theorem of composition of functions for holomorphic functional calculus given by:

Proposition 2.4

Let $\Omega _{1}$ and $\Omega _{2}\$be two open subsets of $\mathbb {C}$. Let $f\in \mathcal {H}\left( \Omega _{1},A\right)$ and $g\in \mathcal {H} \left( \Omega _{2},A\right)$ satisfy the condition that for every compact set $K_{2}$ in $\Omega _{2}$, there exists a compact $K_{1}$ in $\Omega _{1}$ such that $Sp_{A}\left( g(z)\right)$ is contained in $K_{1}$, for every z in $K_{2}$. Let $a\mathbf {\in }A$ with $Sp_{A}(a\mathbf {)\subset }\Omega _{2}$ and $Sp_{A}\left( g(a)\right) \subset \Omega _{1}$. Then $f\circ g(a)=f(g(a)).$

Proof

Let $\Gamma _{2}$ be a positively oriented simple closed rectifiable contour such that $Sp_{A}(a)\mathbf {\subset }int(\Gamma _{2})$ and $int(\Gamma _{2})\cup \Gamma _{2}\subset \Omega _{2}$. By our assumption, there exists a compact set $K_{1}$ in $\Omega _{1}$ such that $Sp_{A}\left( g(z)\right) \subset K_{1}$ for all $z\in int(\Gamma _{2})\cup \Gamma _{2}$. Now, since $int(\Gamma _{2})\cup \Gamma _{2}$ is a compact subset of $\Omega _{2}$ ([3], Definition 3.1, p.45), one can choose a suitable positively oriented simple closed rectifiable contour $\Gamma _{1}$ so that both $Sp_{A}\left( g(a)\right)$ and $K_{1}$ are contained in $int(\Gamma _{1})$ and $int(\Gamma _{1})\cup \Gamma _{1}\subset \Omega _{1}$. Then in the same manner as before, one has:

$$\begin{aligned} f\left( g(a)\right) =\frac{1}{2\pi i}\int _{\Gamma _{1}}f(w)\left( we-g(a)\right) ^{-1}dw. \end{aligned}$$

By 2) of Remark 2.3, one has:

$$\begin{aligned} \left( we-g(a)\right) ^{-1}=\frac{1}{2\pi i}\int _{\Gamma _{2}}\left( we-g(z)\right) ^{-1}(ze-a)^{-1}dz. \end{aligned}$$

It follows that:

$$\begin{aligned} f\left( g(a)\right) =\left( \frac{1}{2\pi i}\right) ^{2}\int _{\Gamma _{1}}\int _{\Gamma _{2}}f(w)\left( we-g(z)\right) ^{-1}(ze-a)^{-1}dzdw. \end{aligned}$$

The continuity of in $\left( w,z\right)$ on $\Gamma _{1}\times \Gamma _{2}$ ([2], Proposition 6, p. 11) of the function:

$$\begin{aligned} \left( w,z\right) \longmapsto f(w)\left( we-g(z)\right) ^{-1}(ze-a)^{-1} \end{aligned}$$

allows us to change the order of integration. Thus

$$\begin{aligned} f\left( g(a)\right) =\left( \frac{1}{2\pi i}\right) ^{2}\int _{\Gamma _{2}} \left[ \int _{\Gamma _{1}}f(w)\left( we-g(z)\right) ^{-1}dw\right] (ze-a)^{-1}dz. \end{aligned}$$

Now, by (3),

$$\begin{aligned} f\left( g(z)\right) =\frac{1}{2\pi i}\int _{\Gamma _{1}}f(w)\left( we-g(z)\right) ^{-1}dw. \end{aligned}$$

Whence

$$\begin{aligned} f\left( g(a)\right)= & {} \frac{1}{2\pi i}\int _{\Gamma _{2}}f(g(z))(ze-a)^{-1}dz \\= & {} f\circ g(a). \end{aligned}$$

This completes the proof. $\square$

Now, we examine one of the most powerful properties of holomorphic functional calculus. That is the spectral mapping theorem:

Proposition 2.5

Let A be a complex unital Banach algebra and $\textbf{a}$ be a c.f.e and $\Omega$ as described in Definition 2.1. Then the mapping $\Phi _{\textbf{a }}$ defined by (2) is an algebra homomorphism of $\mathcal {H}\left( \Omega ,A\right)$ into $A\$such that

$$\begin{aligned} \widehat{\Phi _{\textbf{a}}\left( f\right) }\left( \chi \right) =\left( \chi \circ f\right) \left( \chi (\mathbf {a)}\right) )\text {, for every }\chi \in Sp(A)\text {.} \end{aligned}$$

Proof

Let $\chi \in Sp(A)$. Then$\ \widehat{\Phi _{\textbf{a}}\left( f\right) } \left( \chi \right) =\chi \left( \Phi _{\textbf{a}}\left( f\right) \right)$ and

$$\begin{aligned} \chi \left( \Phi _{\textbf{a}}\left( f\right) \right)= & {} \frac{1}{\left( 2\pi i\right) ^{n}}\int _{T^{n}\left( z^{0},r\right) }\chi \left( f(z)C( \textbf{a},z)\right) dz \\= & {} \frac{1}{\left( 2\pi i\right) ^{n}}\int _{T^{n}\left( z^{0},r\right) }\chi \left( f(z)\right) \chi \left( C(\textbf{a},z)\right) dz. \end{aligned}$$

Therefore, taking into account the fact that $\chi \left( C(\textbf{a},z)\right) =\left( C(\chi \left( \textbf{a}\right) ,z)\right)$, we have

$$\begin{aligned} \chi \left( \Phi _{\textbf{a}}\left( f\right) \right)= & {} \frac{1}{\left( 2\pi i\right) ^{n}}\int _{T^{n}\left( z^{0},r\right) }\left( \chi \circ f\right) \left( z\right) \left( C(\chi \left( \textbf{a}\right) ,z)\right) dz \\= & {} \left( \chi \circ f\right) \left( \chi (\mathbf {a)}\right) ). \end{aligned}$$

$\square$

Proposition 2.6

Let A be a complex unital Banach algebra and $\Omega$ be an open subset of $\mathbb {C}^{n}$ as described in Definition 2.1. If $f\in \mathcal {H} \left( \Omega ,A\right)$ and $\textbf{a}=\left( a_{1},\ldots ,a_{n}\right) \in A^{n}$ be a c.f.e. with $Sp(\mathbf {a)\subset }D^{n}\left( z^{0},r\right)$, then

(1)

$$\begin{aligned} Sp_{A}\left( f(\mathbf {a)}\right) \mathbf {\subset }\bigcup \limits _{\lambda \in Sp(\mathbf {a)}}Sp_{A}\left( f(\lambda )\right) . \end{aligned}$$

(2) If $f=\widetilde{f}e$, where $\widetilde{f}$ is a holomorphic scalar function on $\Omega$, then

$$\begin{aligned} Sp_{A}\left( f(\mathbf {a)}\right) \mathbf {=}f(Sp(\mathbf {a)}). \end{aligned}$$

Proof

(1) Observe first that if f has no inverse on $Sp(\mathbf {a)}$, then $g=f^{-1}$is holomorphic in an open set $\Omega _{1}$ such that $Sp( \mathbf {a)\subset }\Omega _{1}\subset \Omega$. Since $fg=e$ in $\Omega _{1}$, it follows that $f\left( \textbf{a}\right) g\left( \textbf{a}\right) =e$ and $f(\textbf{a})$ is invertible. Now fix $\beta \in \mathbb {C}$. Then $\beta \in Sp_{A}\left( f(\mathbf {a)}\right)$ if and only if $f(\mathbf {a)-} \beta e$ is not invertible in A. From the above, there exists $\lambda \in Sp(\mathbf {a)}$ such that $f(\lambda \mathbf {)-}\beta e$ is not invertible in A, that is $\beta \in Sp_{A}\left( f(\lambda \mathbf {)}\right) \mathbf {. }$ (2) If $f=\widetilde{f}e$, where $\widetilde{f}$ is a holomorphic scalar function on $\Omega$, then $\bigcup \nolimits _{\lambda \in Sp\left( \textbf{a}\right) }Sp\left( f(\lambda )\right) =f(Sp(\mathbf {a))}.$ Furthermore, for $\chi \in Sp(A)$, one has

$$\begin{aligned} \widehat{f(\mathbf {a)}}\left( \chi \right) =\left( \chi \circ f\right) \left( \chi (\mathbf {a)}\right) )=f\left( \chi (\mathbf {a)}\right) . \end{aligned}$$

Therefore $f(Sp(\mathbf {a))\subset }Sp_{A}\left( f(\mathbf {a)}\right) \mathbf {.}$ $\square$

Remark 2.7

The inclusion 1) of Proposition 2.6. can be strict as the simple example shows. Let $x\in A$ be an invertible element such that $Sp_{A}\left( x\right) =\left\{ 1,2\right\}$ and put $f(z)=x^{-1}z$. Since $f(x)=e,$ we have $Sp_{A}\left( f(x\mathbf {)}\right) \mathbf {=}\left\{ 1\right\}$ but $Sp_{A}\left( f(1\mathbf {)}\right) \mathbf {=}\left\{ \frac{1}{2},1\right\} .$

3 A vector-valued n-hamonic functional calculus

In this section we define a functional calculus for A-valued n-harmonic functions of several variables and describe some of its properties. Let $z\in D^{n}\left( z^{0},r\right)$ and $w\in T^{n}\left( z^{0},r\right)$. Then the Poisson kernel $P^{n}\left( z,w\right)$ is the product

$$\begin{aligned} P^{n}\left( z,w\right) =P\left( z_{1},w_{1}\right) \ldots P\left( z_{n},w_{n}\right) , \end{aligned}$$

where $P\left( z_{i},w_{i}\right)$ is the classical Poisson kernel for the disk $D\left( z_{i}^{0},r_{i}\right)$. Note that

$$\begin{aligned} P\left( z_{i},w_{i}\right)= & {} Re\left[ \left( w_{i}+z_{i}-2z_{i}^{0}\right) \left( w_{i}-z_{i}\right) ^{-1}\right] \\= & {} \left( \overline{w_{i}}-\overline{z_{i}}\right) ^{-1}\left[ r_{i}^{2}-\left( \overline{z_{i}}-\overline{z_{i}^{0}}\right) \left( z_{i}-z_{i}^{0}\right) \right] \left( w_{i}-z_{i}\right) ^{-1} \\= & {} \overline{C\left( z_{i},w_{i}\right) }\left[ r_{i}^{2}-\left( \overline{ z_{i}}-\overline{z_{i}^{0}}\right) \left( z_{i}-z_{i}^{0}\right) \right] C\left( z_{i},w_{i}\right) . \end{aligned}$$

If we put

$$\begin{aligned} \Delta \left( z_{i},z_{i}^{0}\right) =r_{i}^{2}-\left( \overline{z_{i}}- \overline{z_{i}^{0}}\right) \left( z_{i}-z_{i}^{0}\right) , \end{aligned}$$

then

$$\begin{aligned} P\left( z_{i},w_{i}\right) =\overline{C\left( z_{i},w_{i}\right) }\ \Delta \left( z_{i},z_{i}^{0}\right) C\left( z_{i},w_{i}\right) \end{aligned}$$

and

$$\begin{aligned} P\left( z,w\right) =\overline{C\left( z,w\right) }\ \prod \limits _{i=1}^{n}\Delta \left( z_{i},z_{i}^{0}\right) C\left( z,w\right) . \end{aligned}$$

We also put

$$\begin{aligned} \Delta ^{n}\left( z,z^{0}\right) =\prod \limits _{i=1}^{n}\Delta \left( z_{i},z_{i}^{0}\right) \end{aligned}$$

Let A be a complex unital Banach algebra with continuous involution $x\longmapsto x^{*}$, $z^{0}=\left( z_{1}^{0},\ldots ,z_{n}^{0}\right) \in \mathbb {C}^{n}$, $r=\left( r_{1},\ldots ,r_{n}\right) \in \mathbb {R}_{+}^{n}$. If $\textbf{x}=\left( x_{1},\ldots ,x_{n}\right) \in A^{n}$ be a c.f.e. with $Sp_{A}(\mathbf {x)\subset }D^{n}\left( z^{0},r\right)$, then the A-valued Poisson kernel is defined by the equality:

$$\begin{aligned} P\left( \textbf{x},w\right) =C\left( \textbf{x},w\right) ^{*}\Delta ^{n}\left( \textbf{x},z^{0}\right) C\left( \textbf{x},w\right) ,\ w\in T^{n}\left( z^{0},r\right) , \end{aligned}$$

where

$$\begin{aligned} \Delta ^{n}\left( \textbf{x},z^{0}\right) =\prod \limits _{i=1}^{n}\Delta \left( x_{i},z_{i}^{0}\right) =\prod \limits _{i=1}^{n}\left[ r_{i}^{2}-\left( x_{i}^{*}-\overline{z_{i}^{0}}\right) \left( x_{i}-z_{i}^{0}\right) \right] . \end{aligned}$$

If $n=1$, then

$$\begin{aligned} P\left( x_{1},w\right)= & {} C\left( x_{1},w\right) ^{*}\Delta ^{1}\left( x_{1},z^{0}\right) C\left( x_{1},w\right) \\= & {} \left( w-x_{1}^{*}\right) ^{-1}\left[ r^{2}-\left( x_{1}^{*}- \overline{z_{1}^{0}}\right) \left( x_{1}-z^{0}\right) \right] \left( w-x_{1}\right) ^{-1} \\= & {} Re\left[ \left( w+x_{1}-2z_{1}^{0}\right) \left( w-x_{1}\right) ^{-1}) \right] \ge 0. \end{aligned}$$

Definition 3.1

Let A be a complex unital Banach algebra with continuous involution $x\longmapsto x^{*}$, $\Omega$ an open subset of $\mathbb {C}^{n}$, $z^{0}=\left( z_{1}^{0},\ldots ,z_{n}^{0}\right)$ $\in \Omega$,$\ r=\left( r_{1},\ldots ,r_{n}\right) \in \mathbb {R}_{+}^{n}$ such that $\overline{D}^{n}\left( z^{0},r\right) \subset \Omega ,$ $\textbf{x }=\left( x_{1},\ldots ,x_{n}\right) \in A^{n}$ be a c.f.e. with $Sp(\textbf{x}) \mathbf {\subset }D^{n}\left( z^{0},r\right)$ and $f\in h(\Omega ,A).$ Then the element of A given by the Poisson integral formula:

$$\begin{aligned} \frac{1}{\left( 2\pi \right) ^{n}}\int _{T^{n}\left( z^{0},r\right) }f(w)P( \textbf{x},w)\frac{\left| dw_{1}\right| }{r_{1}}\ldots \frac{\left| dw_{n}\right| }{r_{n}} \end{aligned}$$

is denoted by $P\left[ f\right] (\textbf{x})$.

If we denote by $\Psi _{\textbf{x}}(f)$ or just $f(\textbf{x})$ the element $P\left[ f\right] (\textbf{x})$, one has a mapping of $h(\Omega ,A)$ into A, noted $\Psi _{\textbf{x}}$, given by:

$$\begin{aligned} \Psi _{\textbf{x}}:h(\Omega ,A)\longrightarrow A:f\longmapsto \Psi _{\textbf{ x}}\left( f\right) =P\left[ f\right] (\textbf{x})=f(\textbf{x}). \end{aligned}$$

Then $\Psi _{\textbf{x}}$ is an involutive homomorphism from $h(\Omega ,A)$ into A that extends the algebra homomorphism $\Theta _{\textbf{a}}$ given by the Cauchy transform ([18], Proposition 9, p. 103). Furthermore, if $K\$ is a compact neighbourhood contained in $\Omega$ and containing $Sp(\textbf{ x})$, then the mapping $\Psi _{\textbf{x}}$ is continuous with respect to the uniform convergence on K.

Proposition 3.2

Let A be a hermitian Banach algebra with continuous involution $x\longmapsto x^{*}$, $\Omega$ and $\textbf{x}$ as described in Definition 3.1. If $\textbf{x}$ is normal, then, for every $f\in h\left( \Omega ,A\right) ,$ one has

$$\begin{aligned} \widehat{\Psi _{\textbf{x}}\left( f\right) }\left( \chi \right) =\left( \chi \circ f\right) \left( \chi (\mathbf {x)}\right) )\text {, for every }\chi \in Sp(A). \end{aligned}$$

Corollary 3.3

Let A be a hermitian Banach algebra with continuous involution $x\longmapsto x^{*}$, $\Omega$ and $\textbf{x}$ as described in Definition 3.1. If $\textbf{x}$ is normal, then

(1) $Sp_{A}\left( f(\mathbf {x)}\right) \mathbf {\subset } \bigcup \nolimits _{\lambda \in Sp(\mathbf {x)}}Sp_{A}\left( f(\lambda )\right)$, for every $f\in h\left( \Omega ,A\right) .$

(2) If $f=\widetilde{f}e$, where $\widetilde{f}$ is a harmonic scalar function on $\Omega$, then

$$\begin{aligned} Sp_{A}\left( f(\mathbf {x)}\right) \mathbf {=}f(Sp(\textbf{x})). \end{aligned}$$

Remark 3.4

The proofs of Proposition 3.2 and Corollary 3.3 are similar to those of Proposition 2.5 and Proposition 2.6. Here the hypothesis that $A\$is hermitian is used to get

$$\begin{aligned} \chi \left( P(\textbf{x},w)\right) =P(\chi \left( \textbf{x}\right) ,w)\text { , for every }\chi \in Sp(A) \end{aligned}$$

which results from the fact that every character $\chi$ of a hermitian algebra $A\$is hermitian ([4], (i), Theorem 1.4.1, p. 11), i.e.,

$$\begin{aligned} \chi \left( x^{*}\right) =\overline{\chi \left( x\right) }\text {, for every }x\in A. \end{aligned}$$

4 Some applications

In this section, we give some applications of functional calculi as explored in the preceding sections. Its applications concern a generalization of von Neumann’s theorem ([5], Théorème 6, p. 506), N. Wiener and P. Lé vy theorems ([9], Theorem 4.2, p. 337 and Theorem 5.1, p. 339) and ([6], Theorem 3.1 and Theoerem 3.2). We obtain an analog of Neumann’s theorem for A-valued holomorphic functions of several variables. Afterward we use weighted algebras analogues of the classical theorems of N. Wiener and P. L évy on absolutely Fourier series and we get multi-dimensional versions of N. Wiener and P. Lévy theorems given in ([8], Theorem l, 347 and Theorem 2, p. 349).

4.1 Analog of von Neumann’s theorem.

The spectral inequality of von Neumann (cf. [10], Theorem 1, p. 276) is well-known. Its asserts that, given a contraction T on a Hilbert space $\mathcal {H}$, i.e., $\left\| T\right\| \le 1$ and a complex function f analytic on the open unit disk D. If $f(D)\subset D$, then f(T) is also a contraction on $\mathcal {H}$. In ([7], Theorem 3.1, p. 933), the third author showed that hermitian algebras are the natural framework of the last inequality. He also obtained an extension to analytic A-valued functions ([5], Théorème 1, p. 498). Here, we obtain a generalization of the von-Neumann’s inequality to several variables.

In the sequel, A will denote a hermitian Banach algebra with continuous involution $x\longmapsto x^{*}$ and $D^{n}=D^{n}\left( 0,1\right)$. We consider:

$$\begin{aligned} \mathcal {H}N_{A}(D^{n})= & {} \left\{ f\in \mathcal {H}(D^{n},A):f(z)\text { is normal, for every }z\in D^{n}\right\} \\ \mathcal {H}_{A}(D^{n})= & {} \left\{ \begin{array}{c} f\in \mathcal {H}N_{A}(D^{n}):f(z)f(w)=f(w)f(z)\text {,} \\ \text {for every }z,w\in D^{n} \end{array} \right\} \\ B_{A}\left( D^{n}\right)= & {} \left\{ f\in \mathcal {H}N_{A}(D^{n}):\left| f(z)\right| <1\text {, for every }z\in D^{n}\right\} \\ P_{A}\left( D^{n}\right)= & {} \left\{ g\in \mathcal {H}N_{A}(D^{n}):Reg(z)>0 \text {, for every }z\in D^{n}\right\} , \end{aligned}$$

where Reg(z) designates the real part of g(z).

As a first application of the n-harmonic functional calculus, we have the following result:

Theorem 4.1

Let $\mathbf {a=}\left( a_{1},\ldots ,a_{n}\right) \mathbf {\in }A^{n}$ be a c.f.e. such that $\left| a_{i}\right| <1,$ for every $i=1,\ldots ,n.$ If $P(\textbf{a},w)>0$ for every w in the torus $T^{n}(0,1)$, then $Reg( \textbf{a})>0$, with g in$\ P_{A}\left( D^{n}\right)$.

Proof

Since $g\in P_{A}\left( D^{n}\right)$, one has $g\in \mathcal {H} N_{A}(D^{n})\$and its real part Reg is an A-valued harmonic function on $D^{n}$. Let $\mathbf {a\in }A^{n}$ be a c.f.e. such that $\left| a_{i}\right| <1,$ for every $i=1,\ldots n$. Choose positive numbers $r_{i}\$and$\ r_{i}^{\prime }$ with $\left| a_{i}\right|<r_{i}<r_{i}^{\prime }<1$. It is easy to verify that $Sp_{A}(\mathbf { a)\subset }D^{n}\left( 0,r\right)$, where $r=\left( r_{1},\ldots ,r_{n}\right)$. By hypothesis, g in$\ P_{A}\left( D^{n}\right)$. It follows that $Reg(z)>0$, for every $z\in \overline{D}^{n}\left( 0,r^{\prime }\right)$, i.e., Reg(z) is a positive and invertible element of A. Consider the function

$$\begin{aligned} \psi (z)=\rho \left( \left[ Reg(z)\right] ^{-1}\right) \text {, for every } z\in \overline{D}^{n}\left( 0,r^{\prime }\right) . \end{aligned}$$

As the spectral radius $x\longmapsto \rho (x)$ is upper-semicontinuous on A, the function $\psi$ is therefore upper semicontinuous on $\overline{D} ^{n}\left( 0,r^{\prime }\right)$.So $\psi$ has a maximum on $\overline{ D}^{n}\left( 0,r^{\prime }\right)$. Therefore, there exists $\delta >0$ such that:

$$\begin{aligned} \rho \left( \left[ Reg(z)\right] ^{-1}\right) \le \dfrac{1}{\delta },\text { for every }z\in \overline{D}^{n}\left( 0,r^{\prime }\right) . \end{aligned}$$

Whence $\rho \left( Reg(z)\right) >\delta$, for every $z\in \overline{D} ^{n}\left( 0,r^{\prime }\right)$. It follows that $Reg(z)>\delta$, for every $z\in D^{n}\left( 0,r^{\prime }\right)$. Consider h defined by:

$$\begin{aligned} h(z)=Reg(z)-\delta e\text {, for every }z\in D^{n}\left( 0,r^{\prime }\right) . \end{aligned}$$

Thus, by Definition 3.1, one has:

$$\begin{aligned} h(\textbf{a})=\frac{1}{\left( 2\pi \right) ^{n}}\int _{T^{n}\left( 0,r\right) }h(w)P(\textbf{a},w)\frac{\left| dw_{1}\right| }{r_{1}}\ldots \frac{ \left| dw_{n}\right| }{r_{n}}. \end{aligned}$$

By our assumption, $P(\textbf{a},w)\ge 0$, for every $w\in T^{n}(0,1).\$ Then, since $\textbf{a}$ and $\mathcal {H}(D^{n},A)$ are commuting, we have

$$\begin{aligned} h(w)P(\textbf{a},w)\ge 0,\text { for every }w\in T^{n}(0,r). \end{aligned}$$

Indeed, for a fixed $w\in T^{n}(0,r)$, one has $h(w)>0$ and $P(\textbf{a},w)\ge 0.$ Thus there exists $u,v\in H(A)$ such that

$$\begin{aligned} h(w)=u^{2}\text { and }P(\textbf{a},w)=v^{2}. \end{aligned}$$

Moreover u and v commutes since h and $\textbf{a}$ are commuting. It follows that $h(w)P(\textbf{a},w)\in H(A)$ and

$$\begin{aligned} h(w)P(\textbf{a},w)=u^{2}v^{2}=uv\left( uv\right) ^{*}\ge 0\text { by }(1). \end{aligned}$$

So $h(\textbf{a})\ge 0.$ Finally, since $h(\textbf{a})=Reg(\textbf{a} )-\delta$, we have $Reg(\textbf{a})-\delta \ge 0$, i.e., $Sp_{A}(Reg( \textbf{a})-\delta )\subset [ 0,+\infty [$. Whence $Sp_{A}\left( Reg(\textbf{a})\right) \subset [ \delta ,+\infty [.$ Thus $Reg(\textbf{a})$ is a hermitian element of $A\$and $Sp_{A}\left( Reg( \textbf{a})\right) \subset ] 0,\infty [$ for $\delta >0.$ So $Reg(\textbf{a})>0.$ This completes the proof. $\square$

As in the complex case, the reader can prove that the relations

$$\begin{aligned} g(z)=\left( e+f(z)\right) \left( e-f(z)\right) ^{-1}\ \text {and}\ \ f(z)=(g(z)-e)(g(z)+e)^{-1} \end{aligned}$$

establish a bijection between the functions f in $B_{A}\left( D^{n}\right) \$and the functions g in $P_{A}\left( D^{n}\right)$. Using this fact, we obtain an equivalent version of Theorem 4.1 given by:

Theorem 4.2

Let $f\in B_{A}\left( D^{n}\right)$ and $\mathbf {a=}\left( a_{1},\ldots ,a_{n}\right) \mathbf {\in }A^{n}$ be a c.f.e. such that $\left| a_{i}\right| <1,$ for every $i=1,\ldots ,n.$ If $P(\textbf{a},w)>0$, for every $w\in T^{n}(0,1)$, then $\left| f(\textbf{a} )\right| <1$.

Remark 4.3

(1) In the case where $n=1$, we have

$$\begin{aligned} P\left( \textbf{a},w\right)= & {} Re\left[ \left( w+a_{1}\right) \left( \left( w-a_{1}\right) ^{-1}\right) \right] \\= & {} \left( \overline{w}-a_{1}^{*}\right) ^{-1}\left( 1-a_{1}^{*}a_{1}\right) \left( w-a_{1}\right) ^{-1}. \end{aligned}$$

As $\left| a_{1}\right| <1$, we have $e-a_{1}^{*}a_{1}>0$, so that

$$\begin{aligned} e-a_{1}^{*}a_{1}=u^{2}\ \text { for some } \ u\in H(A). \end{aligned}$$

Hence

$$\begin{aligned} P\left( a_{1},w\right) =\left( \overline{w}-a_{1}^{*}\right) ^{-1}u\ u\ \left( w-a_{1}\right) ^{-1}. \end{aligned}$$

Then, by (1), we have $P\left( a_{1},w\right) \ge 0.$

(2) In the case where $n=2$ and $\mathbf {a=}\left( a_{1},a_{2}\right)$, we have:

$$\begin{aligned} P^{2}\left( \textbf{a},w\right) =P\left( a_{1},w_{1}\right) P\left( a_{2},w_{2}\right) \end{aligned}$$

Put $P\left( a_{1},w_{1}\right) =h^{2}$ and $P\left( a_{2},w_{2}\right) =k^{2}$, then $P^{2}\left( \textbf{a},w\right) =h^{2}k^{2}$ and as

$$\begin{aligned} Sp_{A}\left( h^{2}k^{2}\right) =Sp_{A}\left( khhk\right) \text {,} \end{aligned}$$

we obtain $Sp_{A}\left[ P^{2}\left( \textbf{a},w\right) \right] \subset \mathbb {R}^{+}.$

Remark 4.4

Using Theorem 4.1, we obtain as in [5], the analog of Schawrz’s lemma ([5], Théorème 4, p. 502) as well as the analog of Pick’s theorem ([5], Théorème 5, p. 504).

4.2 Analogues of Lévy and Wiener’s theorems

For $p\in ] 1,+\infty [$, let $\omega :\mathbb {Z} ^{k}\longrightarrow [ 1,+\infty [$, $k\in \mathbb {N}^{*}\$ fixed, be a weight on $\mathbb {Z}^{k},$ i.e., $\omega$ satisfies

$$\begin{aligned} \underset{m\in \mathbb {Z}^{k}}{\sum }\omega ^{\frac{1}{1-p}}(m)<+\infty . \end{aligned}$$

(4)

For $n=\left( n_{1},\ldots ,n_{k}\right) \in \mathbb {Z}^{k}$ and $t=\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}$, we will use the notation $(n,t)=n_{1}t_{1}+\cdots +n_{k}t_{k}$. Now, we consider the following weighted space:

$$\begin{aligned} \mathcal {A}_{k}^{p}\left( \omega \right) =\left\{ f:\mathbb {R} ^{k}\longrightarrow \mathbb {C}:f(t)=\sum \limits _{n\in \mathbb {Z} ^{k}}a_{n}e^{i(n,t)}:\left( a_{n}\right) _{n\in \mathbb {Z}^{k}}\in l_{\omega }^{p}\left( \mathbb {Z}^{k}\right) \right\} , \end{aligned}$$

where $l_{\omega }^{p}\left( \mathbb {Z}^{k}\right)$ stands for the space of all sequences $\left( a_{n}\right) _{n\in \mathbb {Z}}$ with $a_{n}\in \mathbb {C}$ and

$$\begin{aligned} \sum _{n\in \mathbb {Z}^{k}}\left| a_{n}\right| ^{p}\omega \left( n\right) <+\infty . \end{aligned}$$

In $l_{\omega }^{p}\left( \mathbb {Z}^{k}\right)$, we introduce convolution multiplication given by:

$$\begin{aligned} a*b=\left\{ \sum _{i\in \mathbb {Z}}a_{i}b_{n-i}\right\} _{n} \end{aligned}$$

and we suppose that there exists a constant $\gamma =\gamma \left( \omega \right) >0$ such that:

$$\begin{aligned} \omega ^{\frac{1}{1-p}}*\omega ^{\frac{1}{1-p}}\le \gamma \omega ^{ \frac{1}{1-p}}. \end{aligned}$$

(5)

Then $l_{\omega }^{p}\left( \mathbb {Z}^{k}\right)$ becomes a Banach algebra ([6], Theorem 3.3). The space $\mathcal {A}_{k}^{p}\left( \omega \right)$ endowed with the norm $\left\| .\right\| _{k,p,\omega }$ defined by:

$$\begin{aligned} \left\| f\right\| _{k,p,\omega }=\left( \sum _{n\in \mathbb {Z} ^{k}}\left| a_{n}\right| ^{p}\omega \left( n\right) \right) ^{\frac{1 }{p}}\text {, for every }f\in \mathcal {A}_{k}^{p}\left( \omega \right) \text {,} \end{aligned}$$

and with the classical pointwise multiplication, becomes a Banach algebra. In the sequel, we suppose:

$$\begin{aligned} \underset{\left| n\right| \longrightarrow +\infty }{\lim }\left( \omega \left( n\right) \right) ^{\frac{1}{n_{j}}}=1, \ \text { for every }\ j=1,\ldots k. \end{aligned}$$

(6)

where $\left| n\right| =n_{1}+\cdots +n_{k},$ denotes the length of $n=\left( n_{1},\ldots ,n_{k}\right) \in \mathbb {Z}^{k}$ and

$$\begin{aligned} \omega \left( n+m\right) \le \omega \left( n\right) \omega \left( m\right) \text {, for every }n,m\in \mathbb {Z}^{k} \end{aligned}$$

(7)

Recall that every character of the algebra $\mathcal {A}_{k}^{p}\left( \omega \right)$ is an evaluation at some $t^{0}\in \mathbb {R}^{k}$ ([6], Theorem 3.3), where $t^{0}=\left( t_{1}^{0},\ldots ,t_{k}^{0}\right)$ with $0\le t_{j}^{0}<2\pi$, for every $j=1,\ldots ,k$, and so,

$$\begin{aligned} Sp\left( \mathcal {A}_{k}^{p}\left( \omega \right) \right) =\left\{ \chi _{t}:t\in [ 0,2\pi [ ^{k}\right\} , \end{aligned}$$

where $\chi _{t}(f)=f(t)$, for every $f\in \mathcal {A}_{k}^{p}\left( \omega \right)$, and

$$\begin{aligned} Sp(f)=\left\{ f(t):t\in [ 0,2\pi [ ^{k}\right\} . \end{aligned}$$

Also the Jacobson radical of $\mathcal {A}_{k}^{p}\left( \omega \right)$, denoted by $Rad\left( \mathcal {A}_{k}^{p}\left( \omega \right) \right) ,$ is:

$$\begin{aligned} Rad\left( \mathcal {A}_{k}^{p}\left( \omega \right) \right) =\bigcap \limits _{\chi \in Sp\left( \mathcal {A}_{k}^{p}\left( \omega \right) \right) }\ker \chi . \end{aligned}$$

Whence $\mathcal {A}_{k}^{p}\left( \omega \right)$ is semi-simple, i.e., $Rad\left( \mathcal {A}_{k}^{p}\left( \omega \right) \right) =\left\{ 0\right\}$.

Using the fact that the spectrum of an element f of the algebra $\mathcal {A }_{1}^{p}\left( \omega \right)$ is nothing other than the set of values of f, we obtain the following generalization of P. Lévy theorem for holomorphic functions of several variables.

Theorem 4.5

(Multi-dimensional holomorphic version of P. Lévy theorem) Let $p\in ] 1,+\infty [$ and $\omega$ be a weight on $\mathbb {Z}$ satisfying (5), (6) and (7). Let $f=\left( f_{1},\ldots ,f_{k}\right)$, where $f_{j}(t)=\sum \limits _{n\in \mathbb {Z} }a_{n,j}e^{int}$, where $\left( a_{n,j}\right) _{n\in \mathbb {Z}}\subset \mathbb {C}$, for $j=1,\ldots ,k$, is a periodic function such that:

$$\begin{aligned} \left\| f_{j}\right\| _{p,\omega }=\left( \sum _{n\in \mathbb {Z} }\left| a_{n,j}\right| ^{p}\omega \left( n\right) \right) ^{\frac{1}{ p}}<+\infty \text {.} \end{aligned}$$

Let $\Omega$ be an open subset of $\mathbb {C}^{k}$ containing the image of the function f. Let $F\in \mathcal {H}(\Omega ,\mathcal {A} _{1}^{p}\left( \omega \right) )$. Then F(f) also can be developed in a trigonometric series $F(f)(t)=\sum \limits _{n\in \mathbb {Z}}b_{n}e^{int}$, where $\left( b_{n}\right) _{n\in \mathbb {Z}}\subset \mathbb {C}$, such that:

$$\begin{aligned} \left\| F(f)\right\| _{p,\omega }=\left( \sum \limits _{n\in \mathbb {Z} }\left| b_{n}\right| ^{p}\omega \left( n\right) \right) ^{\frac{1}{p} }<+\infty \end{aligned}$$

and, for every $t\in \mathbb {R}$,

$$\begin{aligned} F(f)(t)=F\left( f_{1}(t),\ldots ,f_{k}(t)\right) \left( t\right) =\sum \limits _{n\in \mathbb {Z}}b_{n}e^{int}. \end{aligned}$$

If moreover $F\$is a holomorphic scalar function on $\Omega$, then, for every $t\in \mathbb {R}$,

$$\begin{aligned} F\left( f_{1}(t),\ldots ,f_{k}(t)\right) =\sum \limits _{n\in \mathbb {Z} }b_{n}e^{int}. \end{aligned}$$

Now we consider, in the algebra $\mathcal {A}_{k}^{p}\left( \omega \right)$, the algebra involution $f\longmapsto f^{*}$ defined by:

$$\begin{aligned} f^{*}(t)=\sum \limits _{n\in \mathbb {Z}^{k}}\overline{a_{-n}}e^{i\left( n,t\right) }\text {, for every }t\in \mathbb {R}^{k}. \end{aligned}$$

Since the algebra $\mathcal {A}_{k}^{p}\left( \omega \right)$ is semi-simple, the involution is continuous ([2], Theorem 2, p.191). Moreover $\left( \mathcal {A}_{k}^{p}\left( \omega \right) ,\left\| .\right\| _{p,\omega }\right)$ is a hermitian algebra.

4.3 Another generalization of Wiener and Lévy theorems

We will now consider complex functions of several variables and analytic functional calculus for a single variable to give generalization of N. Wiener and P. Lévy theorems.

As an immediate consequence, we obtain the following multi-dimen-sional generalization of the N. Wiener theorem.

Theorem 4.6

(Multi-dimensional generalization of N. Wiener theorem) Let $p\in ] 1,+\infty [$ and $\omega$ be a weight on $\mathbb {Z} ^{k}$ satisfying (5), (6) and (7). Let $f(t)=f(t_{1},\ldots ,t_{k})$ be a $2\pi$-periodic function with respect to each variable, represented by a series

$$\begin{aligned} f(t)=\sum \limits _{n\in \mathbb {Z}^{k}}a_{n}e^{i(n,t)} \end{aligned}$$

such that

$$\begin{aligned} \left\| f\right\| _{k,p,\omega }=\left( \sum _{n\in \mathbb {Z} ^{k}}\left| a_{n}\right| ^{p}\omega \left( n\right) \right) ^{\frac{1 }{p}}<+\infty . \end{aligned}$$

If f(t) is invertible, for every $t\in \mathbb {R}^{k}$, then the function $f^{-1}$ can be developed in a trigonometric series $f^{-1}\left( t\right) =\sum \limits _{n\in \mathbb {Z}^{k}}b_{n}e^{i(n,t)}$, where $\left( b_{n}\right) _{n}$ is a sequence in $\mathcal {A}_{k}^{p}\left( \omega \right) ,$ such that:

$$\begin{aligned} \left\| f^{-1}\right\| _{k,p,\omega }=\left( \sum _{n\in \mathbb {Z} ^{k}}\left| b_{n}\right| ^{p}\omega \left( n\right) \right) ^{\frac{1 }{p}}<+\infty . \end{aligned}$$

Using holomorphic functional calculus and Theorem 2.2, we also obtain as a consequence, the following multi-dimensional generalization of the Lévy theorem.

Theorem 4.7

(Multi-dimensional generalization of P. Lévy theorem) Let $p\in ] 1,+\infty [$ and $\omega$ be a weight on $\mathbb {Z} ^{k}$ satisfying (5), (6) and (7). Let $f(t)=f(t_{1},\ldots ,t_{k})$ be a $2\pi$-periodic $\mathcal {A}_{k}^{p}\left( \omega \right)$-valued function with respect to each variable, represented by a series $f(t)=\sum \limits _{n\in \mathbb {Z}^{k}}a_{n}e^{i(n,t)}$, such that

$$\begin{aligned} \left\| f\right\| _{k,p,\omega }=\left( \sum _{n\in \mathbb {Z} ^{k}}\left| a_{n}\right| ^{p}\omega \left( n\right) \right) ^{\frac{1 }{p}}<+\infty . \end{aligned}$$

Let $\Omega$ be an open subset of $\mathbb {C}^{n}$, $z^{0}=\left( z_{1}^{0},\ldots ,z_{n}^{0}\right) \in \Omega$,$\ r=\left( r_{1},\ldots ,r_{n}\right) \in \mathbb {R}_{+}^{n}$ such that $\overline{D} ^{n}\left( z^{0},r\right) \subset \Omega ,$ and $f\left( \mathbb {R} ^{k}\right) \subset D^{n}\left( z^{0},r\right) \mathbf {.}$ If$\ F\in h(\Omega ,\mathcal {A}_{k}^{p}\left( \omega \right) ),$ then

$$\begin{aligned} P\left[ F\right] (f)=\frac{1}{\left( 2\pi \right) ^{n}}\int _{T^{n}\left( z^{0},r\right) }F(w)P(f,w)\frac{\left| dw_{1}\right| }{r_{1}}\ldots \frac{\left| dw_{n}\right| }{r_{n}}. \end{aligned}$$

can be developed in a trigonometric series

$$\begin{aligned} P\left[ F\right] (f)(t)=F(f(t))(t)=\sum \limits _{n\in \mathbb {Z} ^{k}}b_{n}e^{i(n,t}), \end{aligned}$$

such that:

$$\begin{aligned} \left\| P\left[ F\right] (f)\right\| _{k,p,\omega }=\left( \sum _{n\in \mathbb {Z}^{k}}\left| b_{n}\right| ^{p}\omega \left( n\right) \right) ^{\frac{1}{p}}<+\infty . \end{aligned}$$

If moreover $F\in h(\Omega ,\mathbb {C)}$ is an n-harmonic scalar function on $\Omega$, then, for every $t\in \mathbb {R}^{k}$,

$$\begin{aligned} P\left[ F\right] (f)(t)=F(f(t))=\sum \limits _{n\in \mathbb {Z} ^{k}}b_{n}e^{i(n,t)}. \end{aligned}$$

Remark 4.8

Under the assumptions of the Theorem 4.7, if$\ F\in h\left( \Omega ,\mathcal { A}_{k}^{p}\left( \omega \right) \right) ,$ then, by Proposition 3.2, we have:

$$\begin{aligned} \widehat{P\left[ F\right] \left( f\right) }\left( \chi \right) =\left( \chi \circ F\right) \left( \chi (f\mathbf {)}\right) )\text {, for every }\chi \in Sp(A). \end{aligned}$$

(8)

This implies that:

$$\begin{aligned} P\left[ F\right] (f)(t)=F(f(t))(t)\text {, for every }t\in \mathbb {R}^{k}. \end{aligned}$$

While if $F\in h(\Omega ,\mathbb {C)}$, then (8) becomes as follows:

$$\begin{aligned} \widehat{P\left[ F\right] \left( f\right) }\left( \chi \right) =F\left( \chi (f\mathbf {)}\right) )\text {, for every }\chi \in Sp(A). \end{aligned}$$

So, one has

$$\begin{aligned} P\left[ F\right] (f)(t)=F(f(t))\text {, for every }t\in \mathbb {R}^{k}.\ \end{aligned}$$