A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

Duren, Peter; Gallardo-Gutiérrez, Eva A.; Montes-Rodrı́guez, Alfonso

doi:10.1112/blms/bdm026

Cited by 60 publications

(42 citation statements)

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“…A special case of this problem is considered, for example, in [13]. It is perhaps surprising that this does not always hold; for example, Theorem 2 in [10] makes exactly this natural mistake.…”

Section: Convolution Operators 121mentioning

confidence: 94%

Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

Harper

2008

Complex Anal. Oper. Theory

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We ask when convolution operators with scalar-or operator-valued kernel functions map between weighted L 2 spaces of Hilbert space-valued functions. For a certain class of decreasing weights, including negative powers (t + a) −m for example, we solve the one-weight problem completely by using Laplace transforms and Bergman-type spaces of vector-valued analytic functions. For a much more general class of decreasing weights, we solve the one-weight problem for all positive real kernels (also for L p (w) with p > 1), by results on Steklov operators which generalise the weighted Hardy inequality. When the kernel function is a strongly continuous semigroup of bounded linear Hilbert space operators, which arises from input-output maps of certain linear systems, then the most obvious sufficient condition for boundedness, obtained by taking norm signs inside the integrals, is also necessary in many cases, but not in general. Mathematics Subject Classification (2000). Primary 44A35; Secondary 47B32, 47D06.

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Section: Convolution Operators 121mentioning

confidence: 94%

Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

Harper

2008

Complex Anal. Oper. Theory

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“…A Paley-Wiener Theorem for A 2 (C + ) states that the Laplace transform L is an isometric isomorphism between L 2 (R + , t −1 dt) and A 2 (C + ) (see [1], for instance).…”

Section: Example 25mentioning

confidence: 99%

“…B Jonathan R. Partington J.R.Partington@leeds.ac.uk 1 require the operator to have more than one point in its spectrum (and satisfy other conditions which involves the growth of the resolvent). Even in the case of compact operators, the proof of the existence of invariant subspaces reduces to the quasinilpotent case (non-quasinilpotent compact operators have, obviously, eigenspaces).…”

Section: Introductionmentioning

confidence: 99%

A continuous model for quasinilpotent operators

2016

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A classical result due to Foias and Pearcy establishes a discrete model for every quasinilpotent operator acting on a separable, infinite-dimensional complex Hilbert space H. More precisely, given a quasinilpotent operator T on H, there exists a compact quasinilpotent operator K in H such that T is similar to a part of K ⊕ K ⊕· · ·⊕ K ⊕· · · acting on the direct sum of countably many copies of H. We show that a continuous model for any quasinilpotent operator can be provided. The consequences of such a model will be discussed in the context of C 0 -semigroups of quasinilpotent operators.

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“…We will use the definition given in [3,Section 6] for the weighted Dirichlet spaces on Π + : For all α > −1, the space D 2 α (Π + ) consists of the analytic functions F :…”

Section: The Spectrum Ofmentioning

confidence: 99%

Spectra of linear fractional composition operators on the Hardy and weighted Bergman spaces of the half-plane

Schroderus

2017

Journal of Mathematical Analysis and Applications

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Abstract. We compute the spectra and the essential spectra of bounded linear fractional composition operators acting on the Hardy and weighted Bergman spaces of the upper halfplane. We are also able to extend the results to weighted Dirichlet spaces of the upper halfplane. IntroductionThe Boundedness of a composition operator C τ on the Hardy or the weighted Bergman spaces of the half-plane has been proved to be equivalent with the angular derivative of the inducing map at infinity, denoted by τ ′ (∞), being finite and positive (see [14,6]). In addition, Elliott et al. [6,5] have shown that whenever C τ is bounded, the operator norm, the essential operator norm and the spectral radius are all equal and determined by the quantity τ ′ (∞). The above properties show that the spaces H 2 (Π + ) and A 2 α (Π + ) differ significantly from their unit disc analogues H 2 (D) and A 2 α (D) regarding the composition operators acting on them. We refer e.g. to [21,2] for expositions of the rich theory of composition operators on spaces defined on the unit disc.It is natural to ask what the spectral properties of composition operators acting on the halfplane are. In fact, in the unit disc setting the spectral picture of composition operators has been completely determined when the inducing maps are linear fractional transformations and the operators act on weighted Dirichlet spaces, including H 2 (D) and A 2 α (D) for all α > −1 (see, for instance, [17,2,10,8,19,7]). The spectra in the corresponding setting on the half-plane are largely unknown; in the (unweighted) Dirichlet space of Π + the spectra are known but besides that only the spectra (and the essential spectra) for invertible or self-adjoint parabolic and invertible hyperbolic composition operators acting on the Hardy space H 2 (Π + ) have been computed (see [15]). In contrast to the unit disc case, not all linear fractional transformations τ induce bounded composition operators on. Indeed, C τ is bounded only when τ is a parabolic or a hyperbolic self-map of Π + fixing infinity. In this paper we compute the spectra and the essential spectra of these composition operators.In the parabolic case (see Theorem 3.1 in Section 3) we obtain the following result:Theorem A. Let τ be a parabolic self-map of Π + , that is, τ (w) = w + w 0 , where Im w 0 ≥ 0 and w 0 = 0. Then the spectrum of C τ acting on the Hardy or the weighted Bergman spaces of the upper half-plane equals2010 Mathematics Subject Classification. 47B33. Key words and phrases. Composition operator, spectrum, essential spectrum, Hardy space of the upper halfplane, weighted Bergman spaces of the upper half-plane.The author is supported by the Magnus Ehrnrooth Foundation in Finland.

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A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

Cited by 60 publications

References 4 publications

Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

A continuous model for quasinilpotent operators

Spectra of linear fractional composition operators on the Hardy and weighted Bergman spaces of the half-plane

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