Unified Approach to Spectral Properties of Multipliers
Abstract: Let Bn be the open unit ball in C n . We characterize the spectra of pointwise multipliers Mu acting on Banach spaces of analytic functions on Bn satisfying some general conditions. These spaces include Bergman-Sobolev spaces A p α,β , Bloch-type spaces Bα, weighted Hardy spaces H p w with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces H 2 β . Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is… Show more
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 1 publication
References 18 publications
Cyclicity in the Drury-Arveson space and other weighted Besov spaces
Let H \mathcal {H} be a space of analytic functions on the unit ball B d {\mathbb {B}_d} in C d \mathbb {C}^d with multiplier algebra M u l t ( H ) \mathrm {Mult}(\mathcal {H}) . A function f ∈ H f\in \mathcal {H} is called cyclic if the set [ f ] [f] , the closure of { φ f : φ ∈ M u l t ( H ) } \{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\} , equals H \mathcal {H} . For multipliers we also consider a weakened form of the cyclicity concept. Namely for n ∈ N 0 n\in \mathbb {N}_0 we consider the classes C n ( H ) = { φ ∈ M u l t ( H ) : φ ≠ 0 , [ φ n ] = [ φ n + 1 ] } . \begin{equation*} \mathcal {C}_n(\mathcal {H})=\{\varphi \in \mathrm {Mult}(\mathcal {H}):\varphi \ne 0, [\varphi ^n]=[\varphi ^{n+1}]\}. \end{equation*} Many of our results hold for N N :th order radially weighted Besov spaces on B d {\mathbb {B}_d} , H = B ω N \mathcal {H}= B^N_\omega , but we describe our results only for the Drury-Arveson space H d 2 H^2_d here. Letting C s t a b l e [ z ] \mathbb {C}_{stable}[z] denote the stable polynomials for B d {\mathbb {B}_d} , i.e. the d d -variable complex polynomials without zeros in B d {\mathbb {B}_d} , we show that a m p ; if d is odd, then C s t a b l e [ z ] ⊆ C d − 1 2 ( H d 2 ) , and a m p ; if d is even, then C s t a b l e [ z ] ⊆ C d 2 − 1 ( H d 2 ) . \begin{align*} &\text { if } d \text { is odd, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d-1}{2}}(H^2_d), \text { and }\\ &\text { if } d \text { is even, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d}{2}-1}(H^2_d). \end{align*} For d = 2 d=2 and d = 4 d=4 these inclusions are the best possible, but in general we can only show that if 0 ≤ n ≤ d 4 − 1 0\le n\le \frac {d}{4}-1 , then C s t a b l e [ z ] ⊈ C n ( H d 2 ) \mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d) . For functions other than polynomials we show that if f , g ∈ H d 2 f,g\in H^2_d such that f / g ∈ H ∞ f/g\in H^\infty and f f is cyclic, then g g is cyclic. We use this to prove that if f , g f,g extend to be analytic in a neighborhood of B d ¯ \overline {{\mathbb {B}_d}} , have no zeros in B d {\mathbb {B}_d} , and the same zero sets on the boundary, then f f is cyclic in ∈ H d 2 \in H^2_d if and only if g g is. Furthermore, if the boundary zero set of f ∈ H d 2 ∩ C ( B d ¯ ) f\in H^2_d\cap C(\overline {{\mathbb {B}_d}}) embeds a cube of real dimension ≥ 3 \ge 3 , then f f is not cyclic in the Drury-Arveson space.
Cyclicity in the Drury-Arveson space and other weighted Besov spaces
Let H \mathcal {H} be a space of analytic functions on the unit ball B d {\mathbb {B}_d} in C d \mathbb {C}^d with multiplier algebra M u l t ( H ) \mathrm {Mult}(\mathcal {H}) . A function f ∈ H f\in \mathcal {H} is called cyclic if the set [ f ] [f] , the closure of { φ f : φ ∈ M u l t ( H ) } \{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\} , equals H \mathcal {H} . For multipliers we also consider a weakened form of the cyclicity concept. Namely for n ∈ N 0 n\in \mathbb {N}_0 we consider the classes C n ( H ) = { φ ∈ M u l t ( H ) : φ ≠ 0 , [ φ n ] = [ φ n + 1 ] } . \begin{equation*} \mathcal {C}_n(\mathcal {H})=\{\varphi \in \mathrm {Mult}(\mathcal {H}):\varphi \ne 0, [\varphi ^n]=[\varphi ^{n+1}]\}. \end{equation*} Many of our results hold for N N :th order radially weighted Besov spaces on B d {\mathbb {B}_d} , H = B ω N \mathcal {H}= B^N_\omega , but we describe our results only for the Drury-Arveson space H d 2 H^2_d here. Letting C s t a b l e [ z ] \mathbb {C}_{stable}[z] denote the stable polynomials for B d {\mathbb {B}_d} , i.e. the d d -variable complex polynomials without zeros in B d {\mathbb {B}_d} , we show that a m p ; if d is odd, then C s t a b l e [ z ] ⊆ C d − 1 2 ( H d 2 ) , and a m p ; if d is even, then C s t a b l e [ z ] ⊆ C d 2 − 1 ( H d 2 ) . \begin{align*} &\text { if } d \text { is odd, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d-1}{2}}(H^2_d), \text { and }\\ &\text { if } d \text { is even, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d}{2}-1}(H^2_d). \end{align*} For d = 2 d=2 and d = 4 d=4 these inclusions are the best possible, but in general we can only show that if 0 ≤ n ≤ d 4 − 1 0\le n\le \frac {d}{4}-1 , then C s t a b l e [ z ] ⊈ C n ( H d 2 ) \mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d) . For functions other than polynomials we show that if f , g ∈ H d 2 f,g\in H^2_d such that f / g ∈ H ∞ f/g\in H^\infty and f f is cyclic, then g g is cyclic. We use this to prove that if f , g f,g extend to be analytic in a neighborhood of B d ¯ \overline {{\mathbb {B}_d}} , have no zeros in B d {\mathbb {B}_d} , and the same zero sets on the boundary, then f f is cyclic in ∈ H d 2 \in H^2_d if and only if g g is. Furthermore, if the boundary zero set of f ∈ H d 2 ∩ C ( B d ¯ ) f\in H^2_d\cap C(\overline {{\mathbb {B}_d}}) embeds a cube of real dimension ≥ 3 \ge 3 , then f f is not cyclic in the Drury-Arveson space.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
