Hilbert Space Of Analytic Functions Research Articles

Ahlfors problem for polynomials

We present a conjecture that the asymptotics for Chebyshev polynomials in a complex domain can be given in terms of the reproducing kernels of a suitable Hilbert space of analytic functions in this domain. It is based on two classical results due to Garabedian and Widom. To support this conjecture we study the asymptotics for Ahlfors extremal polynomials in the complement to a system of intervals on , arcs on , and the asymptotics of the extremal entire functions for the continuous counterpart of this problem. Bibliography: 35 titles.

Complex symmetric operators and interpolation

In this note we study the interpolation properties of complex symmetric operators on Hilbert spaces. We show that under certain conditions the complex symmetricity property is preserved under quadratic interpolation. We apply this result to the study of complex symmetric Toeplitz operators on weighted Hilbert spaces of analytic functions.

Kreĭn space representation and Lorentz groups of analytic Hilbert modules

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk $H^2({\mathbb~D}^2)$. A closed subspace $M$ in $H^2({\mathbb~D}^2)$ is called asubmodule if $z_{i}M\subset~M$ $(i=1,~2)$. An associated integral operator (em defect operator $C_{M}$ captures much information about $M$. Using a Kreuin space indefinite metric on the range of $C_{M}$, this paper gives a representation of $M$. Then it studies the group (called Lorentz group) of isometric self-maps of $M$ with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for $M$.

On Some Properties of Cowen-Douglas Class of Operators

We will consider multiplication operators on a Hilbert space of analytic functions on a domainΩ⊂C. For a bounded analytic functionφonΩ, we will give necessary and sufficient conditions under which the complement of the essential spectrum ofMφinφΩbecomes nonempty and this gives conditions for the adjoint of the multiplication operatorMφbelongs to the Cowen-Douglas class of operators. Also, we characterize the structure of the essential spectrum of a multiplication operator and we determine the commutants of certain multiplication operators. Finally, we investigate the reflexivity of a Cowen-Douglas class operator.

Sarason’s Toeplitz product problem on the Fock–Sobolev space

The Sarason’s Toeplitz product problem asks when the Toeplitz product operator T u T v , with analytic symbols u and v, is bounded on Hilbert space of analytic functions. In this paper, we deal with this problem on the Fock–Sobolev space and have a complete solution that u = e q , v = Ce −q , where q is a linear complex polynomial and C is a nonzero constant.

Generalized Composition Operators on Weighted Hilbert Spaces of Analytic Functions

Letφbe an analytic self-map of the open unit disk D andgbe an analytic function on D. The generalized composition operator induced by the mapsgandφis defined by the integral operatorI(g,φ)f(z) =∫0zf′(φ(ς))g(ς)dς. Given an admissible weightω, the weighted Hilbert spaceHωconsists of all analytic functionsfsuch that ∥f∥2Hω= |f(0)|2+∫D|f′(z)|2ω(z)dA(z) is finite. In this paper, we characterize the boundedness and compactness of the generalized composition operators on the spaceHωusing theω-Carleson measures. Moreover, we give a lower bound for the essential norm of these operators.

Sarason's Toeplitz product problem for a class of Fock spaces

Sarason's Toeplitz product problem asks when the operator TuTv‾ is bounded on various Hilbert spaces of analytic functions, where u and v are analytic. The problem is highly nontrivial for Toeplitz operators on the Hardy space and the Bergman space (even in the case of the unit disk). In this paper, we provide a complete solution to the problem for a class of Fock spaces on the complex plane. In particular, this generalizes an earlier result of Cho, Park, and Zhu.

Szegő–Widom asymptotics of Chebyshev polynomials on circular arcs

Thiran and Detaille give an explicit formula for the asymptotics of the sup-norm of the Chebyshev polynomials on a circular arc. We give the so-called Szegő–Widom asymptotics for this domain, i.e., explicit expressions for the asymptotics of the corresponding extremal polynomials. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on this arc. That is, we give explicit formulas for the asymptotics of the error of approximation as well as of the extremal functions. Our computations show that in the proper normalization the limit of the upper envelope represents the diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. Due to Garabedian, the analytic capacity in an arbitrary domain is the diagonal of the corresponding Szegő kernel. We do not know any result of this kind with respect to upper envelopes of polynomials. If this is a general fact or a specific property of the given domain, we rise as an open question.

The Spectrum of a Composition Operator with Automorphic Symbol

We investigate the spectrum of a composition operator with automorphic symbol on a variety of Banach and Hilbert spaces of analytic functions. In particular, the spectrum is determined when the operator acts on the little Bloch space, VMOA, and a family of weighed Dirichlet spaces.

Multipliers of Hilbert spaces of analytic functions on the complex half-plane

It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted $L^2$ functions defined on $(0, \infty)$ and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic functions are not only Hilbert spaces but also Banach algebras, and are therefore contained within their spaces of multipliers. We provide some necessary as well as sufficient conditions for this to happen and look at its consequences.

Multipliers of Hilbert spaces of analytic functions on the complex half-plane

It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted $L^2$ functions defined on $(0, \infty)$ and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic functions are not only Hilbert spaces but also Banach algebras, and are therefore contained within their spaces of multipliers. We provide some necessary as well as sufficient conditions for this to happen and look at its consequences.

Carleson measures for Hilbert spaces of analytic functions on the complex half-plane

The notion of a Carleson measure was introduced by Lennart Carleson in his proof of the Corona Theorem for H∞(D). In this paper we will define it for certain type of reproducing kernel Hilbert spaces of analytic functions of the complex half-plane, C+, which will include Hardy, Bergman and Dirichlet spaces. We will obtain several necessary or sufficient conditions for a positive Borel measure to be Carleson by preforming tests on reproducing kernels, weighted Bergman kernels, and studying the tree model obtained from a decomposition of the complex half-plane. The Dirichlet space will be investigated in detail as a special case. Finally, we will present a control theory application of Carleson measures in determining admissibility of controls in well-posed linear evolution equations.

Partial orders on partial isometries

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBrangesRovnyak spaces. This characterization is applied to investigate properties of these pre-orders and the equivalence classes they generate.

A remark on the multipliers on spaces of Weak Products of functions

AbstractIf H denotes a Hilbert space of analytic functions on a region Ω ⊆ Cd , then the weak product is defined byWe prove that if H is a first order holomorphic Besov Hilbert space on the unit ball of Cd , then the multiplier algebras of H and of H ⊙ H coincide.

DYNAMICS OF THE DILATION WEIGHTED COMPOSITION OPERATORS

In this paper we investigate the hypercyclicity of adjoint of a special weighted composition operators on Hilbert spaces of analytic functions on the open unit disc. AMS Subject Classification: 47B37, 47B38

Adjoint of some composition operators on the Dirichlet and Bergman spaces

Let $\varphi$ be a holomorphic self-map of the unit disk $\mathbb{U}:=\{z\in \mathbb{C}: |z| < 1\}$, and the composition operator with symbol $\varphi$ is defined by $C_\varphi f=f \circ \varphi.$ In this paper we present formula for the adjoint of composition operators in some Hilbert spaces of analytic functions, in the case that $\varphi$ is a finite Blaschke product or a rational univalent holomorphic self-map of the unit disk $\mathbb{U}$.

Linear graph transformations on spaces of analytic functions

Let H be a Hilbert space of analytic functions with multiplier algebra M(H), and letM={(f,T1f,…,Tn−1f):f∈D} be an invariant graph subspace for M(H)(n). Here n≥2, D⊆H is a vector-subspace, Ti:D→H are linear transformations that commute with each multiplication operator Mφ∈M(H), and M is closed in H(n). In this paper we investigate the existence of non-trivial common invariant subspaces of operator algebras of the typeAM={A∈B(H):AD⊆D:ATif=TiAf∀f∈D}. In particular, for the Bergman space La2 we exhibit examples of invariant graph subspaces of fiber dimension 2 such that AM does not have any nontrivial invariant subspaces that are defined by linear relations of the graph transformations for M.

Adjoints of generalized composition operators with rational symbol

Given $\varphi:\mathbb{U}\rightarrow\mathbb{U}$, an analytic self-map of the open unit disc in complex plane, the composition operator $C_{\varphi}$ is defined by $C_{\varphi}f=f\circ\varphi$ for $f$ belonging to some Hilbert space of analytic functions on $\mathbb{U}$. In the present paper, we introduce a generalization of the composition operators and reproducing kernel functions on the weighted Hardy spaces. We also obtain the adjoints of generalized composition operators with rational symbol acting on the Hardy, Bergman and Dirichlet spaces.

Hypercyclic Behavior of Translation Operators on Spaces of Analytic Functions on Hilbert Spaces

We consider special Hilbert spaces of analytic functions of many infinite variables and examine composition operators on these spaces. In particular, we prove that under some conditions a translation operator is bounded and hypercyclic.

Quantum models with spectrum generated by the flows of polynomial zeros

A class of purely bosonic models is characterized having the following properties in a Hilbert space of analytic functions: (i) wave function is the generating function for orthogonal polynomials of a discrete energy variable ϵ, (ii) any Hamiltonian has nondegenerate purely point spectrum that corresponds to infinite discrete support of measure in the orthogonality relation of the polynomials , (iii) the support is determined exclusively by the points of discontinuity of , (iv) the spectrum of can be numerically determined as fixed points of monotonic flows of the zeros of orthogonal polynomials , (v) one can compute practically an unlimited number of energy levels (e.g. in double precision). If a model of is exactly solvable, its spectrum can only assume one of four qualitatively different types. The results are applied to spin-boson quantum models that are, at least partially, diagonalizable and have at least single one-dimensional irreducible component in the spin subspace. Examples include the Rabi model and its various generalizations.