Spectra of weighted composition operators on algebras of analytic functions on Banach spaces

Cheng Yuan; Ze-Hua Zhou

Displaying similar documents to “Spectra of weighted composition operators on algebras of analytic functions on Banach spaces”

$^{\infty}$ -vectors and boundedness

Jan Stochel, F. H. Szafraniec (1997)

Annales Polonici Mathematici

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The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its $^{\infty}$ -vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

Bounded point evaluations for multicyclic operators

M. EL Guendafi, M. Mbekhta, E. H. Zerouali (2005)

Banach Center Publications

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Let T be a multicyclic operator defined on some Banach space. Bounded point evaluations and analytic bounded point evaluations for T are defined to generalize the cyclic case. We extend some known results on cyclic operators to the more general setting of multicyclic operators on Banach spaces. In particular we show that if T satisfies Bishop’s property (β), then $ℬ_{a} = ℬ ∖ σ_{a p} (T)$ . We introduce the concept of analytic structures and we link it to different spectral quantities. We apply this concept...

On the norm-closure of the class of hypercyclic operators

Christoph Schmoeger (1997)

Annales Polonici Mathematici

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Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if $f (σ_{W} (T)) \cup z \in ℂ : | z | = 1$ is connected, where $σ_{W} (T)$ denotes the Weyl spectrum of T.

Compact endomorphisms of $H^{\infty} (D)$

Joel Feinstein, Herbert Kamowitz (1999)

Studia Mathematica

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Compact composition operators on $H^{\infty} (G)$ , where G is a region in the complex plane, and the spectra of these operators were described by D. Swanton ( Compact composition operators on B(D), Proc. Amer. Math. Soc. 56 (1976), 152-156). In this short note we characterize all compact endomorphisms, not necessarily those induced by composition operators, on $H^{\infty} (D)$ , where D is the unit disc, and determine their spectra.

On operators with the same local spectra

Aleksandar Torgašev (1998)

Czechoslovak Mathematical Journal

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Let $B (X)$ be the algebra of all bounded linear operators in a complex Banach space $X$ . We consider operators $T_{1}, T_{2} \in B (X)$ satisfying the relation $σ_{T_{1}} (x) = σ_{T_{2}} (x)$ for any vector $x \in X$ , where $σ_{T} (x)$ denotes the local spectrum of $T \in B (X)$ at the point $x \in X$ . We say then that $T_{1}$ and $T_{2}$ have the same local spectra. We prove that then, under some conditions, $T_{1} - T_{2}$ is a quasinilpotent operator, that is $∥ {(T_{1} - T_{2})}^{n} ∥^{1 / n} \to 0$ as $n \to \infty$ . Without these conditions, we describe the operators with the same local spectra only in some particular cases.

Co-analytic, right-invertible operators are supercyclic

Sameer Chavan (2010)

Colloquium Mathematicae

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Let denote a complex, infinite-dimensional, separable Hilbert space, and for any such Hilbert space , let () denote the algebra of bounded linear operators on . We show that for any co-analytic, right-invertible T in (), αT is hypercyclic for every complex α with $| α | > β^{- 1}$ , where $β \equiv i n f_{| | x | | = 1} | | T * x | | > 0$ . In particular, every co-analytic, right-invertible T in () is supercyclic.

Essential norm of the difference of composition operators on Bloch space

Ke-Ben Yang, Ze-Hua Zhou (2010)

Czechoslovak Mathematical Journal

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Let $ϕ$ and $ψ$ be holomorphic self-maps of the unit disk, and denote by $C_{ϕ}$ , $C_{ψ}$ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators $C_{ϕ} - C_{ψ}$ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.

On the multiplication operators on spaces of analytic functions

B. Yousefi, S. Foroutan (2005)

Studia Mathematica

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We consider Hilbert spaces of analytic functions on a plane domain Ω and multiplication operators on such spaces induced by functions from $H^{\infty} (Ω)$ . Recently, K. Zhu has given conditions under which the adjoints of multiplication operators on Hilbert spaces of analytic functions belong to the Cowen-Douglas classes. In this paper, we provide some sufficient conditions which give the converse of the main result obtained by K. Zhu. We also characterize the commutant of certain multiplication operators. ...

Weighted composition operators from Zygmund spaces to Bloch spaces on the unit ball

Yu-Xia Liang, Chang-Jin Wang, Ze-Hua Zhou (2015)

Annales Polonici Mathematici

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Let H() denote the space of all holomorphic functions on the unit ball ⊂ ℂⁿ. Let φ be a holomorphic self-map of and u∈ H(). The weighted composition operator $u C_{φ}$ on H() is defined by $u C_{φ} f (z) = u (z) f (φ (z))$ . We investigate the boundedness and compactness of $u C_{φ}$ induced by u and φ acting from Zygmund spaces to Bloch (or little Bloch) spaces in the unit ball.

The spectral decomposition of weighted shifts and the $A^{p}$ condition

Earl Berkson, T. A. Gillespie (1990)

Colloquium Mathematicae

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Operator positivity and analytic models of commuting tuples of operators

Monojit Bhattacharjee, Jaydeb Sarkar (2016)

Studia Mathematica

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We study analytic models of operators of class $C_{\cdot 0}$ with natural positivity assumptions. In particular, we prove that for an m-hypercontraction $T \in C_{\cdot 0}$ on a Hilbert space , there exist Hilbert spaces and ⁎ and a partially isometric multiplier θ ∈ ℳ (H²(),A²ₘ(⁎)) such that $≅_{θ} = A ² ₘ (⁎) ⊖ θ H ² ()$ and $T ≅ P_{_{θ}} M_{z} |_{_{θ}}$ , where A²ₘ(⁎) is the ⁎-valued weighted Bergman space and H²() is the -valued Hardy space over the unit disc . We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their...

The Bergman projection on weighted spaces: L¹ and Herz spaces

Oscar Blasco, Salvador Pérez-Esteva (2002)

Studia Mathematica

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We find necessary and sufficient conditions on radial weights w on the unit disc so that the Bergman type projections of Forelli-Rudin are bounded on L¹(w) and in the Herz spaces $K_{p}^{q} (w)$ .

Conditions equivalent to C* independence

Shuilin Jin, Li Xu, Qinghua Jiang, Li Li (2012)

Studia Mathematica

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Let and be mutually commuting unital C* subalgebras of (). It is shown that and are C* independent if and only if for all natural numbers n, m, for all n-tuples A = (A₁, ..., Aₙ) of doubly commuting nonzero operators of and m-tuples B = (B₁, ..., Bₘ) of doubly commuting nonzero operators of , $S p (A, B) = S p (A) \times S p (B)$ , where Sp denotes the joint Taylor spectrum.

Weighted norm estimates and $L_{p}$ -spectral independence of linear operators

Peer C. Kunstmann, Hendrik Vogt (2007)

Colloquium Mathematicae

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We investigate the $L_{p}$ -spectrum of linear operators defined consistently on $L_{p} (Ω)$ for p₀ ≤ p ≤ p₁, where (Ω,μ) is an arbitrary σ-finite measure space and 1 ≤ p₀ < p₁ ≤ ∞. We prove p-independence of the $L_{p}$ -spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω,μ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on $L_{p}$ -spectral independence...

New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Flavia Colonna (2013)

Open Mathematics

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Let ψ and φ be analytic functions on the open unit disk $𝔻$ with φ( $𝔻$ ) ⊆ $𝔻$ . We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $𝒟$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also...

Composition operators: $N_{α}$ to the Bloch space to $Q_{β}$

Jie Xiao (2000)

Studia Mathematica

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Let $N_{α}$ ,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and $Q_{β}$ are Möbius invariant, but $N_{α}$ is not. We characterize, in function-theoretic terms, when the composition operator $C_{ϕ} f = f ◦ ϕ$ induced by an analytic self-map ϕ of the unit disk defines an operator $C_{ϕ} : N_{α} \to B$ , $B \to Q_{β}$ , $N_{α} \to Q_{β}$ which is bounded resp. compact.