Exact essential norm of generalized Hilbert matrix operators on classical analytic function spaces

Lindström, Mikael; Miihkinen, Santeri; Norrbo, David

doi:10.1016/j.aim.2022.108598

Cited by 9 publications

(3 citation statements)

References 18 publications

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“…If μ is the Lebesgue measure on [0, 1) the matrix H μ reduces to the classical Hilbert matrix H = (n + k + 1) −1 n,k≥0 , which induces the classical Hilbert operator H which has extensively studied recently (see [1,16,17,19,[32][33][34]).…”

Section: Hilbert-type Operatorsmentioning

confidence: 99%

Operators Induced by Radial Measures Acting on the Dirichlet Space

et al. 2023

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Let $${\mathbb {D}}$$ D be the unit disc in the complex plane. Given a positive finite Borel measure $$\mu $$ μ on the radius [0, 1), we let $$\mu _n$$ μ n denote the n-th moment of $$\mu $$ μ and we deal with the action on spaces of analytic functions in $${\mathbb {D}}$$ D of the operator of Hibert-type $${\mathcal {H}}_\mu $$ H μ and the operator of Cesàro-type $${\mathcal {C}}_\mu $$ C μ which are defined as follows: If f is holomorphic in $${\mathbb {D}}$$ D , $$f(z)=\sum _{n=0}^\infty a_nz^n$$ f ( z ) = ∑ n = 0 ∞ a n z n ($$z\in {\mathbb {D}})$$ z ∈ D ) , then $${\mathcal {H}}_\mu (f)$$ H μ ( f ) is formally defined by $${\mathcal {H}}_\mu (f)(z) = \sum _{n=0}^\infty \left( \sum _{k=0}^\infty \mu _{n+k}a_k\right) z^n$$ H μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n + k a k z n ($$z\in {\mathbb {D}}$$ z ∈ D ) and $${\mathcal {C}}_\mu (f)$$ C μ ( f ) is defined by $$\mathcal C_\mu (f)(z) = \sum _{n=0}^\infty \mu _n\left( \sum _{k=0}^na_k\right) z^n$$ C μ ( f ) ( z ) = ∑ n = 0 ∞ μ n ∑ k = 0 n a k z n ($$z\in {\mathbb {D}}$$ z ∈ D ). These are natural generalizations of the classical Hilbert and Cesàro operators. A good amount of work has been devoted recently to study the action of these operators on distinct spaces of analytic functions in $${\mathbb {D}}$$ D . In this paper we study the action of the operators $${\mathcal {H}}_\mu $$ H μ and $${\mathcal {C}}_\mu $$ C μ on the Dirichlet space $${\mathcal {D}}$$ D and, more generally, on the analytic Besov spaces $$B^p$$ B p ($$1\le p<\infty $$ 1 ≤ p < ∞ ).

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Section: Hilbert-type Operatorsmentioning

confidence: 99%

Operators Induced by Radial Measures Acting on the Dirichlet Space

et al. 2023

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“…In recent years there has been much interest in computing the exact norm of the Hilbert matrix operator on different spaces of analytic functions. We refer the reader to the informative introduction of [124]. This paper also mentions open questions.…”

Section: Hypercyclicity and Mean Ergodicitymentioning

confidence: 99%

Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

Bonet

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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In this survey we report about recent work on weighted Banach spaces of analytic functions on the unit disc and on the whole complex plane defined with sup-norms and operators between them. Results about the solid hull and core of these spaces and distance formulas are reviewed. Differentiation and integration operators, Cesàro and Volterra operators, weighted composition and superposition operators and Toeplitz operators on these spaces are analyzed. Boundedness, compactness, the spectrum, hypercyclicity and (uniform) mean ergodicity of these operators are considered.

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“…It is worth noting that ( 1 j+1 ) j∈N is the moment sequence of the Lebesgure measure on [0, 1), which corresponds to the classical Hilbert matrix H = ((j +k+1) −1 ) j,k∈N . See [11,16,17,18,28,29,31,32,33,34,45] for developments of the Hilbert matrix acting on analytic function spaces.…”

Section: Introductionmentioning

confidence: 99%

Comparison of Difficulty Levels of Examples in High School Mathematics Textbooks

Jiang

Wang

2021

School Mathematics Textbooks in China

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In this paper, we address exponential ergodicity for Lévy driven Langevin dynamics with singular potentials, which can be used to model the time evolution of a molecular system consisting of N particles moving in R d and subject to discontinuous stochastic forces. In particular, our results are applicable to the singular setups concerned with not only the Lennard-Jones-like interaction potentials but also the Coulomb potentials. In addition to Harris' theorem, the approach is based on novel constructions of proper Lyapunov functions (which are completely different from the setting for Langevin dynamics driven by Brownian motions), on invoking the Hörmander theorem for non-local operators and on solving the issue on an approximate controllability of the associated deterministic system as well as on exploiting the time-change idea.

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Exact essential norm of generalized Hilbert matrix operators on classical analytic function spaces

Cited by 9 publications

References 18 publications

Operators Induced by Radial Measures Acting on the Dirichlet Space

Operators Induced by Radial Measures Acting on the Dirichlet Space

Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

Comparison of Difficulty Levels of Examples in High School Mathematics Textbooks

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