Abstract
We consider the space \(C^{\left( n\right) }\left( \Omega \right) ,\) the Banach space of continuous functions with n derivatives and the n th derivative continuous in \({\overline{\Omega }},\) where \(\Omega \subset {\mathbb {C}}\) is a starlike region with respect to \(\alpha \in \Omega .\) We use the so-called \(\alpha\)-Duhamel product
to describe usual \(\underset{\alpha }{*}\)-generators of the Banach algebra \(\left( C^{\left( n\right) }\left( \Omega \right) ,\underset{\alpha }{*}\right) ,\) to estimate \(\left\| \left( I-V_{\alpha }\right) ^{m}\right\|\) and to estimate below the norm \(\left\| \delta _{A} ^{m}\right\| ,\) where \(V_{\alpha }\) is the Volterra integration operator defined by \(V_{\alpha }f\left( z\right) =\int \limits _{\alpha }^{z}f\left( t\right) dt\) and \(\delta _{A}\) is the inner derivation operator defined by \(\delta _{A}\left( X\right) :=\left[ X,A\right] .\) We give a new proof of Aleman-Korenblum theorem in one particular case. Namely, we describe V-invariant subspaces in the Hardy space \(H^{p}\) by using Duhamel product.
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1 Introduction
Let \(\alpha \in {\mathbb {C}}\) be a number. Let \(\Omega \subset {\mathbb {C}}\) be a simply connected bounded region containing the point \(\alpha ,\) which is a star-like region with respect to the point \(z=\alpha ,\) i.e., \(\lambda z+\left( 1-\lambda \right) \alpha \in \Omega\) for every \(z\in \Omega\) and \(\lambda ,\) \(0\le \lambda \le 1.\) We define on \(\Omega\) the Banach space \(C^{\left( n\right) }\left( \Omega \right)\) of all continuous functions on \(\Omega\) with the n th derivative continuous on \({\overline{\Omega }}.\) The space \(C^{\left( n\right) }:=C^{\left( n\right) }\left( \Omega \right)\) is a Banach space equipped with the norm
The \(\alpha\)-convolution and \(\alpha\)-Duhamel product are defined in \(C^{\left( n\right) },\) respectively, by
and
where the integral is taken over the segment joining the points \(\alpha\) and z \(\left( z\in \Omega \right) .\) The \(\alpha\)-integration operator \(V_{\alpha }\) is defined on \(C^{\left( n\right) }\) by \(V_{\alpha }f\left( z\right) := { \int \nolimits _{\alpha }^{z}} f\left( t\right) dt,\) where the integration is performed as above over straight-line segments connecting the points \(\alpha\) and z. Our investigation is motivated by the papers [2, 12, 18, 26, 31], where some properties of Banach algebra \(\left( C^{\left( n\right) }\left[ 0,1\right] ,\underset{\alpha }{\circledast }\right)\) and Volterra integration operator \(V_{\alpha }\) are studied. In the present paper, we describe \(\underset{\alpha }{*}\)-generators of the Banach algebra \(\left( C^{\left( n\right) }\left( \Omega \right) ,\underset{\alpha }{*}\right)\) in terms of \(\alpha\)-Duhamel product (Sect. 2). In Sect. 3, we characterize V-invariant subspaces of \(H^{p}\) by involving the Duhamel product method, and hence we give a new proof of a result of Aleman and Korenblum in their paper [2]. In Sect. 4, we calculate norm of orbits of operator \(I-V_{\alpha }\) on the Hardy space \(H^{2}=H^{2}\left( {\mathbb {D}}\right)\) over the unit disc \(\mathbb {D=}\left\{ z\in {\mathbb {C}}:\left| z\right| <1\right\} ,\) which is related Esterle-Katznelson-Tzafriri theorem for Cesàro bounded operators with single-point spectrum in Hilbert spaces (more detailly, see [44]). In Sect. 4, we also estimate in terms of \(\alpha\)-Duhamel products the norm of orbits of inner-derivation \(\delta _{A}\) on \({\mathcal {B}}\left( C^{\left( n\right) }\left( \Omega \right) \right)\) defined by \(\delta _{A}\left( X\right) :=\left[ X,A\right] .\)
2 The \(\mathop *\nolimits _{\alpha }\)-generators of algebra \(C^{(n)}(\Omega )\)
Recall that for a Banach algebra \({\mathcal {B}}\) the radical \({\mathcal {R}}\) of \({\mathcal {B}}\) is the intersection of the kernel of all (strictly) irreducible representations of \({\mathcal {B}}.\) If \({\mathcal {R}}=\left\{ 0\right\}\), then \({\mathcal {B}}\) is said to be semi-simple and if \({\mathcal {R}}={\mathcal {B}}\), then \({\mathcal {B}}\) is called a radical algebra. Equivalently, \({\mathcal {B}}\) is a radical Banach algebra, if for every element \(b\in {\mathcal {B}}\) the associated multiplication operator \(M_{b}a:=ba\) \(\left( a\in {\mathcal {B}}\right) ,\) is quasinilpotent on \({\mathcal {B}}\), i.e., \(\sigma \left( M_{b}\right) =\left\{ 0\right\} .\)
It is classical that \(\underset{k\rightarrow \infty }{\lim }\left\| f^{\underset{\alpha }{*}k}\right\| ^{1/k}=0,\) and so, the space \(\left( C^{\left( n\right) }\left( \Omega \right) ,\underset{\alpha }{*}\right)\) is a radical Banach algebra with respect to the convolution \(\underset{\alpha }{*}\) defined by formula \(\left( 1\right) ;\) here \(f^{\underset{\alpha }{*}k}:=\underset{k}{\underbrace{f\underset{\alpha }{*}...\underset{\alpha }{*}f}}\) is the \(k^{\text {th}}\) iterated convolution of the function f in \(C^{\left( n\right) }\left( \Omega \right) .\) Clearly, \(\left( f\underset{\alpha }{*}f\right) \left( \alpha \right) =0\) for any \(f\in C^{\left( n\right) }\left( \Omega \right)\). Also,
thus, it is easy to verify that \(\left( f^{\underset{\alpha }{*}k}\right) \left( \alpha \right) =0\) for all \(k=1,2,...\). Therefore, we see that a necessary condition for \(f\in C^{\left( n\right) }\left( \Omega \right)\) to generate \(\left( C^{\left( n\right) }\left( \Omega \right) ,\underset{\alpha }{*}\right) ,\) that is, to yield
is that \(f\left( \alpha \right) \ne 0.\) However, it is not yet known whether this condition is sufficient, even for \(\alpha =0\) (see, for instance, Ginsberg and Newman [14] and Karaev [25]). For more detail, see [12].
In the present section, we study the above stated question for the Banach algebra \(\left( C^{\left( n\right) }\left( \Omega \right) ,\underset{\alpha }{*}\right)\) by proving the following theorem, which reduces this question to the case of the subalgebra
Before stating our result, let us formulate two auxiliary lemmas, the proofs of which are quite similar to the proofs of Lemmas 2.1 and 2.2 of the paper [12], and therefore we omit it.
Lemma 1
\(\left( C^{\left( n\right) }\left( \Omega \right) ,\underset{\alpha }{\circledast }\right)\) is a commutative Banach algebra with the unit element \(f={\mathbf {1}}.\)
Lemma 2
The function \(f\in C^{\left( n\right) }\left( \Omega \right)\) is \(\underset{\alpha }{\circledast }\)-invertible if and only if \(f\left( \alpha \right) \ne 0.\)
Theorem 1
Let \(f\in C^{\left( n\right) }\left( \Omega \right)\) be a function such that \(f\left( \alpha \right) \ne 0.\) Let \(F\left( z\right) = {\textstyle \int \limits _{\alpha }^{z}} f\left( t\right) dt.\) Then f is a \(\underset{\alpha }{*}\)-generator of the algebra \(\left( C^{\left( n\right) }\left( \Omega \right) ,\underset{\alpha }{*}\right)\) if and only if F is a \(\underset{\alpha }{\circledast }\)-generator of the subalgebra \(\left( C_{\alpha }^{\left( n\right) }\left( \Omega \right) ,\underset{\alpha }{\circledast }\right) .\)
Proof
In fact, since \(F\left( z\right) = {\textstyle \int \limits _{\alpha }^{z}} f\left( t\right) dt,\) we obtain for all \(g\in C^{\left( n\right) }\left( \Omega \right)\) that
hence \(D_{\alpha ,F}=C_{\alpha ,f}.\) Therefore \(F\underset{\alpha }{\circledast }f=f\underset{\alpha }{*}f,\) so we have
Thus, by induction we get \(C_{\alpha ,f}^{k}f=D_{\alpha ,F}^{k}f\) for all \(k\ge 0,\) which show that
Hence, by using the fact that
where \(\oplus\) stands for the direct sum of subspaces, we see that
Since \(f\left( \alpha \right) \ne 0,\) by Lemma 2 the \(\alpha\)-Duhamel operator \(D_{\alpha ,f}\) is invertible on the space \(C^{\left( n\right) }\left( \Omega \right) .\) On the other hand, by using that
the assertions of the theorem follow from the invertibility of the operator \(D_{\alpha ,f}\) and the representations (2.1) and (2.2). The theorem is proved. \(\square\)
3 On the lattice of V-invariant subspaces in \(H^{p}\)
Let \({\mathbb {D}}\) denote the open unit disc in the complex plane \({\mathbb {C}}.\) The Hardy space \(H^{p}=H^{p}\left( {\mathbb {D}}\right) ,\) \(1\le p<\infty ,\) consist of all analytic functions on \({\mathbb {D}}\) such that
With this norm \(H^{p}\) is a Banach space when \(1\le p<\infty ,\) while for \(0<p<1\) it is a topological vector space with the translation invariant metric \(d\left( f,g\right) =\left\| f-g\right\| _{H^{p}}^{p},\) \(f,g\in H^{p},\) which is not locally convex. For \(p=+\infty ,\) \(H^{\infty }=H^{\infty }\left( {\mathbb {D}}\right)\) is a Banach algebra with the norm \(\left\| f\right\| _{H^{\infty }}:=\sup \left\{ \left| f\left( z\right) \right| :z\in {\mathbb {D}}\right\} .\)
In this section, we will consider the Volterra integration operator V on the Hardy space \(H^{p}\) \(1\le p<+\infty\) and describe its closed nontrivial invariant subspaces. Also, we calculate the norm of orbits \(\left( I-V_{\alpha }\right) ^{n},\) \(n=1,2,...,\) on \(H^{2}.\)
Note that the lattice of all \(V_{\alpha }\)-invariant subspaces of \(H^{p}\) was described by Donoghue [7] in the case when \(p=2\) and \(\alpha =0.\) Donoghue’s method is pure operator theory, and hardly adapted to other values of p and especially if \(\left| \alpha \right| =1.\) Aleman and Korenblum [2] filled this gap. Their approach is based on classical Borel transforms of complex conjugates of \(H^{p}\)-functions on the unit circle \({\mathbb {T}}=\partial {\mathbb {D}},\) which is the entire function defined by
where \(e_{\lambda }\left( z\right) :=e^{\lambda z}\) and \(dm=\frac{\left| dz\right| }{2\pi }\) is the normalized Lebesgue measure on \({\mathbb {T}} \mathbf {.}\) Following [2], note that in contrast to the meagerness of results on invariant subspaces of Volterra operators in complex domains, the study of their real-variable analogs has a long history and an extensive literature (see, for instance, the survey paper of Nikolski [33]). The description of the invariant subspaces for the classical Volterra integration operator \(V:L^{2}\left[ 0,1\right] \rightarrow L^{2}\left[ 0,1\right] ,\)
is essentially the problem posed in 1938 by Gelfand [13] and first solved by Agmon [1] who showed that all V-invariant subspaces of \(L^{2}\left[ 0,1\right]\) have the form
and hence form a linearly ordered lattice, which means unicellularity of operator V. In the sequel, this result has been extended to a larger class of convolution operators by Kalish [19], Sakhnovich [37], Brodski [5] and Sarason [38] (see also [39, 40] and references therein).
In the following theorem, we give another proof of Aleman-Korenblum theorem [2] in the case when \(\alpha =0.\) Our approach is based on the Duhamel product, which was used early by Nagnibida [32], Wigley [46], Tkachenko [42, 43], Dimovski [6], Raychinov [34] and Karaev [20]. The method of Duhamel products is also used in recent works of Ivanova and Melikhov [8,9,10]. For other applications of Duhamel products method, we refer to the works [4, 6, 11, 15,16,17, 21,22,23,24,25, 28,29,30, 35, 36, 41, 47].
Theorem 2
Let \(V:H^{p}\rightarrow H^{p}\) be an integration operator on the Hardy space \(H^{p}\) \(\left( 1\le p<\infty \right) .\) Then
where
Proof
It is easy to see that \(\mathrm {Lat}\left( V\right) \supset \left\{ E^{\left( n\right) }:n\ge 0\right\} ,\) i.e., \(VE^{\left( n\right) }\subset E^{\left( n\right) }\) for all \(n\ge 0.\) Therefore, it remains only to show that \(\mathrm {Lat}\left( V\right) \subset \left\{ E^{\left( n\right) }:n\ge 0\right\} ,\) that is any V-invariant subspace E has the form \(E^{\left( n\right) }\) for some \(n\ge 0.\) For this aim, we will need the following lemmas. \(\square\)
Lemma 3
([47]). For any two functions \(f,g\in H^{p}\) we have
where \(C>0\) is an absolute constant and
for some constant \(C_{p}>0,\) i.e., \(\left( H^{p},\circledast \right)\) \(\left( 1\le p\le +\infty \right)\) is a Banach algebra.
The proof of the following lemma is contained, for instance, in Wigley’s paper [47].
Lemma 4
Let \(f\in H^{p}\) \(\left( 1\le p\le \infty \right)\) be a nonzero function. Then f is \(\circledast\)-invertible if and only if \(f\left( 0\right) \ne 0.\)
The extreme case \(p=\infty\) included in Wigley’s theorem [47]. So, we prove only the case \(1\le p<+\infty .\) If \(f\in H^{p}\) is \(\circledast\)-invertible, then there is a function \(g\in H^{p}\) such that
From this it is easy to see that \(\left( f\circledast g\right) \left( 0\right) =f\left( 0\right) g\left( 0\right) =1,\) whence \(f\left( 0\right) \ne 0.\) Conversely, if \(f\left( 0\right) \ne 0,\) then we put \(F\left( z\right) :=f\left( z\right) -f\left( 0\right) ,\) and consider the Duhamel operator \(D_{F}:H^{p}\rightarrow H^{p}\) defined by
According to inequality (3.1), it is a bounded operator on \(H^{p}.\) We will show that \(D_{F}\) is even compact. Indeed, since F is an analytic function on the unit disc \({\mathbb {D}},\) we have
where \({\widehat{F}}\left( n\right) =\frac{F^{\left( n\right) }\left( 0\right) }{n!},\) \(n\ge 0.\) We consider the partial sum:
Then
for all \(g\in H^{p}.\) Hence
Since, V is compact, we conclude that \(D_{F_{N}}\) is compact on \(H^{p}\) for any \(N>0.\) So, by (3.1), we have that
for \(1\le p<+\infty .\) Passing to the limit in (3.2) as \(N\rightarrow \infty ,\) we have that \(D_{F}\) is a compact operator.
Now consider operator \(D_{f}\) (with symbol f), and assume that \(g\in \ker \left( D_{F}\right) ,\) that is
Whence \(f\left( 0\right) g\left( 0\right) =0,\) and hence \(g\left( 0\right) =0,\) because \(f\left( \alpha \right) \ne 0.\) Similarly, we get
for all \(z\in {\mathbb {D}},\) and evaluation at 0 gives \(g^{\prime }\left( 0\right) =0.\) By induction, we obtain that \(g^{\left( n\right) }\left( 0\right) =0,\) \(n\ge 1,\) and hence \(g\equiv 0.\) This shows that \(\ker \left( D_{f}\right) =\left\{ 0\right\} .\) Since \(D_{f}=f\left( 0\right) I+D_{F},\) thus we deduce by Fredholm alternative that \(D_{f}\) is invertible in \(H^{p}.\) The lemma is proved.
Recall that the function \(f\in H^{p}\) is a cyclic vector for V if
The set of all cyclic vectors of V is denoted by \(\mathrm {Cyc}\left( V\right) .\)
Lemma 5
Let \(f\in H^{p}\). Then \(f\in \mathrm {Cyc}\left( V\mid E^{\left( n\right) }\right) _{\left( n\ge 0\right) }\) if and only if \(f\in E^{\left( n\right) }\diagdown E^{\left( n+1\right) }\) \(\left( n\ge 0\right)\).
Proof
The proof of the lemma uses some arguments of the paper [39]. Let us introduce the following Duhamel product in the subspace \(E^{\left( n\right) }:\)
Clearly, for \(n=0\), the product \(\overset{0}{\circledast }\) coincides with the usual Duhamel product (1.2).
Let \(f\in E^{\left( n\right) }\) and \(f\notin E^{\left( n+1\right) }\). Formula (3.3) implies that
for each \(k\ge 0\) and \(n=0,1,....\) Expanding function \(f\in E^{\left( n\right) }\) into the Maclaurin series we have that
where \(f^{\left( n\right) }\left( 0\right) \ne 0\) and
Consider the Duhamel operator \(D_{f,n}\) acting in the subspace \(E^{\left( n\right) }\) by the formula \(D_{f,n}g:=f\overset{n}{\circledast }g,\) \(g\in E^{\left( n\right) }\) (see (3.3)). It follows from (3.4) and (3.5) that
The same arguments, as in the proof of Lemma 4, allow us to deduce that \(D_{f,n}\) is invertible in \(E^{\left( n\right) }\), which is omitted. On the other hand, since
according to (3.4), we obtain that
Thus, if \(E^{\left( n\right) }\setminus E^{\left( n+1\right) }\), then \(f\in \mathrm {Cyc}\left( V\mid E^{\left( n\right) }\right)\).
Conversely, the equality \(E_{f}=E^{\left( n\right) }\) implies that \(f^{\left( n+1\right) }\left( 0\right) \ne 0\), and hence \(f\notin E^{\left( n+1\right) }\). Consequently, if \(f\in E^{\left( n\right) }\) and \(f\in \mathrm {Cyc}\left( V\mid E^{\left( n\right) }\right)\), then \(f\notin E^{\left( n+1\right) }\), which proves lemma. Now, we continue the proof of Theorem 2. We will prove that other V-invariant subspaces different from the chain
do not exist, and hence \(\mathrm {Lat}\left( V\right) =\left\{ E^{\left( n\right) }:n=0,1,2,...\right\}\).
In fact, suppose in contrary that there is a nontrivial V-invariant subspace E in \(H^{p}\) which is different from invariant subspaces in (3.6). By virtue of the obvious representation \(E=\cup _{g\in E}E_{g}\) and Lemmas 4 and 5, we see that there exists a function \(f\in E\) such that \(f\left( 0\right) \ne 0\). Therefore, by Lemma 4, we deduce that \(E=H^{p}\), which contradicts to our assumption that \(\left\{ 0\right\} \ne E\ne H^{p}\).
So, according to (3.6), \(\mathrm {Lat}\left( V\right)\) is a linearly ordered set, and hence V is a unicellular operator. The theorem is proven. \(\square\)
4 On orbits of \(I-V_{\alpha }\) and inner derivation
This section is motivated mostly with the papers [31] by Montes, Sanchez and Zemanek and [26] by Leka.
4.1 Norm of orbits of \(I-V_{\alpha }\) on \(H^{2}\)
In this subsection, we calculate the norm of iterates of operator \(I-V_{\alpha }\), where \(V_{\alpha }f= {\displaystyle \int \limits _{\alpha }^{z}} f\left( t\right) dt\) is the Volterra integration operator on \(H^{2}\). Note that the operator \(V=V_{0}\) is the classical Volterra operator with a long history. Many aspects of the Volterra operator has been widely studied and has a vast literature. In particular, Montes, Sanches and Zemanek [31] studied the asymptotic behavior of the powers \(\left( I-V\right) ^{n}\) providing sharp estimates on the norms
Their result gave a negative answer to the question of whether uniform Kreiss boundedness, in general, implies power boundedness under minimal spectral assumption. They also presented sharp estimates on the norms of the differences of consecutive powers of \(I-V\), namely, they obtained that
This also showed that Tsedenbayar’s [45] earlier result in \(L^{2}\left[ 0,1\right]\) was sharp. The main goal of the paper [26] is to provide a closer look at the orbits \(\left( I-V\right) ^{n}f\) when f is in the range of Riemann-Liouville fractional integration operator \(V^{\alpha }\) defined on \(L^{p}\left[ 0,1\right]\) by
where \(\Gamma\) stands for the standard gamma function. More precisely, Leka proved in [26] that
The proof of Leka’s result in [26] is based on exploiting the earlier method of Montes, Sanches and Zemanek in [31] (see also Leka [27]) and Fejer’s asymptotic formula on the Laguerre polynomials. We also note that recently new results and estimates on orbits of operators which are commuting with the Volterra operator have been presented in [3] by Bermudo, Montes and Shkarin. For more details, see [26].
Recall that the Hardy space \(H^{2}=H^{2}\left( {\mathbb {D}}\right)\) is the Hilbert space of analytic functions \(f\left( z\right) = {\displaystyle \sum \limits _{n=0}^{\infty }} a_{n}z^{n}\) such that \(\left\| f\right\| _{2}:=\left( {\displaystyle \sum \limits _{n=0}^{\infty }} \left| a_{n}\right| ^{2}\right) ^{1/2}<+\infty .\)
Proposition 1
Fix an \(\alpha \in {\mathbb {D}}\). Let \(V_{\alpha }\) be the Volterra integration operator on \(H^{2}\) defined by \(V_{\alpha }f\left( z\right) = {\displaystyle \int \limits _{\alpha }^{z}} f\left( t\right) dt\). Then
Proof
The \(\alpha\)-Duhamel product is defined by
It is easy to see from (4.1) that \({\mathbf {1}}\underset{\alpha }{\circledast }f=f\underset{\alpha }{\circledast }{\mathbf {1}}=f\), for all \(f\in H^{2}\) and
The methods of the proofs in [46] and [47] show in particular that \(\left( H^{2},\underset{\alpha }{\circledast }\right)\) is a Banach algebra (with respect to some equivalent norm of \(H^{2}\)). So, it follows from (4.2) that
which implies that \(\left\| V_{\alpha }^{n}\right\| \le \frac{\left\| \left( z-\alpha \right) ^{n}\right\| _{2}}{n!},\) \(\forall n\ge 0\). On the other hand, \(\left\| V_{\alpha }^{n}{\mathbf {1}}\right\| _{2}=\left\| \frac{\left( z-\alpha \right) ^{n}}{n!}\circledast {\mathbf {1}}\right\| _{2}=\frac{\left\| \left( z-\alpha \right) ^{n}\right\| _{2}}{n!}\), and hence,
Similarly, we have
as desired. \(\square\)
Corollary 1
\(\left\| \left( I-V\right) ^{n}\right\| =\left[ {\displaystyle \sum \limits _{k=0}^{n}} \left( \frac{n!}{\left( k!\right) ^{2}\left( n-k\right) !}\right) ^{2}\right] ^{\frac{1}{2}}\).
The following is immediate from Corollary 1.
Corollary 2
\(\left\| \left( I-V\right) ^{n}\right\| \ge \left( {\displaystyle \sum \limits _{k=0}^{n}} \frac{1}{\left( k!\right) ^{4}}\right) ^{\frac{1}{2}}\).
Proposition 2
We have :
The proof is quite similar to the proof Proposition 1, and therefore it is omitted.
4.2 A lower estimate for the norm of orbits of inner derivation operator
We consider the inner derivation operator on \({\mathcal {B}}\left( C^{\left( m\right) }\left( \Omega \right) \right)\) and estimate the norm of its orbit. Let \(A\in {\mathcal {B}}\left( C^{\left( m\right) }\left( \Omega \right) \right)\). The inner derivation operator \(\delta _{A}\) is defined on \({\mathcal {B}}\left( C^{\left( m\right) }\left( \Omega \right) \right)\) by the formula
It is elementary that \(\left\| \delta _{A}\right\| \le 2\left\| A\right\|\). Here, we will prove in terms of \(\alpha\)-Duhamel product a lower estimate for the orbits \(\delta _{A}^{n}\), \(n=2,3,...,\) which improve a result in [18].
Proposition 3
Let \(A\in {\mathcal {B}}\left( C^{\left( m\right) }\left( \Omega \right) \right)\) be fixed. Suppose that for every \(n\ge 1\) and \(X\in {\mathcal {B}} \left( C^{\left( m\right) }\left( \Omega \right) \right)\) there exists \(f_{n,X}\in C^{\left( m\right) }\left( \Omega \right)\) such that
Then
Proof
Since \(\left\| \delta _{A}\right\| \le 2\left\| A\right\|\), the inequality \(\left\| \delta _{A}\right\| \le 4\left\| A\right\|\) is trivial. Further, we have
Hence \(\left\| \delta _{A}^{2}\right\| \le 4\left\| A\right\| ^{2}.\) Similarly,
which implies that \(\left\| \delta _{A}^{3}\right\| \le 4\left\| A\right\| ^{3}\). By induction, we conclude that
as desired.
Now we prove the lower inequality. According to condition, for every \(n\ge 1\) and \(X\in {\mathcal {B}}\left( C^{\left( m\right) }\left( \Omega \right) \right)\) there exists \(f_{n,X}\in C^{\left( m\right) }\left( \Omega \right)\) such that
Denote \(g_{n,X}:=\delta _{A}^{n}\left( X\right) f_{n,X}\). Since \(g_{n,X}\left( \alpha \right) \ne 0\), by Lemma 2, there exists a unique element \(G_{n,X}\in C^{\left( m\right) }\left( \Omega \right)\) such that
Hence, \(f_{n,X}\underset{\alpha }{\circledast }G_{n,X}\underset{\alpha }{\circledast }g_{n,X}=f_{n,X}\). We set
So, it follows from (4.5) that
which implies that \(f_{n,X}\) is an eigenvector of operator \(D_{\alpha ,F_{_{n,X}}}\delta _{A}^{n}\left( X\right)\) corresponding to the eigenvalue \(1\in \sigma _{p}\left( D_{\alpha ,F_{_{n,X}}}\delta _{A}^{n}\left( X\right) \right)\). Then, we obtain that
where \(r\left( .\right)\) denotes the spectral radius of operator. Whence
By taking supremum over the operators \(X\in {\mathcal {B}}\left( C^{\left( m\right) }\left( \Omega \right) \right)\) with \(\left\| X\right\| \le 1\), we have from this inequality that
This proves the proposition. \(\square\)
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Gürdal, M., Garayev, M., Tapdigoglu, R. et al. Some applications of the \(\alpha\)-Duhamel product. J Anal 31, 1557–1572 (2023). https://doi.org/10.1007/s41478-022-00539-2
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DOI: https://doi.org/10.1007/s41478-022-00539-2
Keywords
- \(\alpha\)-Duhamel product
- Starlike region
- Generator
- Inner derivation operator
- Invariant subspace
- Hardy space
- Volterra integration operator