Analytic Hilbert Spaces Research Articles

Hyers–Ulam Stability of the Coefficient Multipliers on Analytic Hilbert Spaces

Hyers–Ulam Stability of the Coefficient Multipliers on Analytic Hilbert Spaces

Approximation numbers of composition operators on weighted besov spaces of analytic functions

AbstractLi et al. [A spectral radius type formula for approximation numbers of composition operators, J. Funct. Anal. 267(12) (2014), 4753-4774] proved a spectral radius type formula for the approximation numbers of composition operators on analytic Hilbert spaces with radial weights and on $H^{p}$ spaces, $p\geq 1$, involving Green capacity. We prove that their formula holds for a wide class of Banach spaces of analytic functions and weights.

Beurling‐Type Theorem for a Class of Reproducing Kernel Hilbert Spaces

In this paper, it is proved that the Beurling‐type theorem holds for the shift operator on a class of reproducing analytic Hilbert spaces.

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers an(Cφ) of composition operators Cφ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ of uniform norm <1, we prove that limn→∞⁡[an(Cφ)]1/n=e−1/Cap[φ(D)], where Cap[φ(D)] is the Green capacity of φ(D) in D. This formula holds also for Hp with 1≤p<∞.

Transitive algebras and reductive algebras on reproducing analytic Hilbert spaces

In this paper, we consider the well-known transitive algebra problem and reductive algebra problem on vector valued reproducing analytic Hilbert spaces. For an analytic Hilbert space H ( k ) with complete Nevanlinna–Pick kernel k, it is shown that both transitive algebra problem and reductive algebra problem on multiplier invariant subspaces of H ( k ) ⊗ C m have positive answer if the algebras contain all analytic multiplication operators. This extends several known results on the problems.

Beurling’s phenomenon on analytic Hilbert spaces over the complex plane

In this paper, we show that Beurling’s theorem on analytic Hilbert spaces over the complex plane analogous to the Hardy space or the Bergman space does not hold, but for finite co-dimensional quasi-invariant subspaces, they are generated by their wandering subspace if and only if they are generated by z n z^n provided that the order of the reproducing kernels K λ ( z ) K_\lambda (z) is less than 2 but not equal to 1.

Analytic Extension of Functions from Analytic Hilbert Spaces*

Let M be an invariant subspace of \( H^{2}_{v} \). It is shown that for each f ∈ M⊥, f can be analytically extended across ∂\( {\partial \mathbb{B}_{d} } \mathord{\left/ {\vphantom {{\partial \mathbb{B}_{d} } {\sigma {\left( {S_{{z_{1} }} , \cdots ,S_{{z_{d} }} } \right)}}}} \right. \kern-\nulldelimiterspace} {\sigma {\left( {S_{{z_{1} }} , \cdots ,S_{{z_{d} }} } \right)}}. \)

Analytic Hilbert Spaces over the Complex Plane

In this paper, we study the structure of quasi-invariant subspaces of analytic Hilbert spaces over the complex plane. We especially investigate when two quasi-invariant subspaces are similar or unitarily equivalent for an analytic Hilbert space over the complex plane.