Hardy Hilbert Space Research Articles

A note on non-supercyclic vectors of Hilbert space operators

In this note it is shown that there is a bounded linear operator T on the Hardy Hilbert space H2 and a vector f in H2 such that the closure of the set {αTnf:α∈ℂ,n≥0} is not H2, but for every subsequence (nk)k=1∞ the closed linear span of {Tnkf:k≥1} is the whole space H2. Furthermore, the closure of {Tng:n≥0} is H2 for some g∈H2.

Berezin number and Berezin norm inequalities for operator matrices

We establish new upper bounds for the Berezin number and Berezin norm of operator matrices, which are refinements of existing bounds. Among other bounds, we prove that if is an operator matrix with for , then and where if i<j and if i>j. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space.

The Toeplitzness of weighted composition operators on Banach spaces of holomorphic functions

ABSTRACT The concepts of various kinds of asymptotically Toeplitz operators have been originally defined and studied for the Hardy–Hilbert space. The aim of this article is to extend these definitions to the case of operators acting on Banach spaces of holomorphic functions, including Hardy spaces, Hardy–Lorentz spaces, and Hardy–Orlicz spaces. We give a function-theoretic characterization of uniformly asymptotically Toeplitzness and mean weakly asymptotically Toeplitzness for weighted composition operators. We also investigate strong and weak asymptotic Toeplitzness of weighted composition operators.

On compositional dynamics on hardy space

In this work, we examine super-recurrence and super-rigidity of composition operators acting on $H(\mathbb{D})$ the space of holomorphic functions on the unit disk $\mathbb{D}$ and on $H^2(\mathbb{D})$ the Hardy-Hilbert space. We characterize the symbols that generate super-recurrent and super-rigid composition operators acting on $H(\mathbb{D})$ and $H^2(\mathbb{D})$.

Compact difference of composition operators on the Hardy spaces

Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of certain Bergman-Carleson measures involving derivatives of the inducing maps. In this paper, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. In particular, our new characterization does not involve derivatives of the inducing maps and includes the Reproducing Kernel Thesis characterization. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.

SPECIAL CONJUGATIONS ON THE HARDY SPACE

A conjugation $C$ on a separable complex Hilbert space $\mathcal H$ is an antilinear operator that is isometric and involutive. In this notes, we characterize all conjugations on the Hardy-Hilbert space $H^{2}$ over the disk. In addition, we characterize complex symmetric Toeplitz operators with a special type of these conjugations.

Chaos for the Dynamics of Toeplitz Operators

Chaotic properties in the dynamics of Toeplitz operators on the Hardy–Hilbert space H2(D) are studied. Based on previous results of Shkarin and Baranov and Lishanskii, a characterization of different versions of chaos formulated in terms of the coefficients of the symbol for the tridiagonal case are obtained. In addition, easily computable sufficient conditions that depend on the coefficients are found for the chaotic behavior of certain Toeplitz operators.

A mean counting function for Dirichlet series and compact composition operators

We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy–Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.

Asymptotics and Isometries of Weighted Composition Operators on Banach Spaces of Holomorphic Functions

We have developed some convergence criteria for the weighted composition operators on various Banach spaces of holomorphic functions, which also extends some of the results of Chalendar and Partington (Weighted Composition Operators: Isometries and Asymptotic Behaviour, 2019). We have also characterized the isometries of the weighted composition operators on the Hardy spaces $$H^p({\mathbb {D}})$$ , which extends the characterization of the isometric weighted composition operators on the Hardy-Hilbert space $$H^2({\mathbb {D}})$$ done in Kumar and Partington (Oper Theory Adv Appl 153:157–167).

Approximability models and optimal system identification

This article considers the problem of optimally recovering stable linear time-invariant systems observed via linear measurements made on their transfer functions. A common modeling assumption is replaced here by the related assumption that the transfer functions belong to a model set described by approximation capabilities. Capitalizing on recent optimal recovery results relative to such approximability models, we construct some optimal algorithms and characterize the optimal performance for the identification and evaluation of transfer functions in the framework of the Hardy Hilbert space and of the disk algebra. In particular, we determine explicitly the optimal recovery performance for frequency measurements taken at equispaced points on an inner circle or on the torus.

Weighted composition operators on the Hilbert Hardy space of a half-plane

ABSTRACTOperators of type acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on the Hilbert Hardy space of a half-plane and study compactness, boundedness, invertibility, normality and spectral properties of such operators.

Some Carleson measures for the Hilbert–Hardy space of tube domains over symmetric cones

In the setting of a general tube domain over a symmetric cone, we obtain a full characterization of measures of the form $$\varphi (y)\,dxdy$$ which are Carleson measures for the Hilbert–Hardy space; for large derivatives, we also obtain a full characterization of general positive measures for which the corresponding embedding operator is continuous. Restricting to the case of the Lorentz cone of dimension three, we prove that by freezing one or two secondary variables, the problem of embedding derivatives of the Hilbert–Hardy space into Lebesgue spaces reduces to the characterization of Carleson measures for Hilbert–Bergman spaces of the upper-half plane or the product of two upper-half planes.

Eigenfunctions of Periodic Differential Operators Analytic in a Strip

Ordinary differential operators with periodic coefficients analytic in a strip act on a Hardy–Hilbert space of analytic functions with inner product defined by integration over a period on the boundary of the strip. Simple examples show that eigenfunctions may form a complete set for a narrow strip, but completeness may be lost for a wide strip. Completeness of the eigenfunctions in the Hardy–Hilbert space is established for regular second order operators with matrix-valued coefficients when the leading coefficient satisfies a positive real part condition throughout the strip.

Quasinilpotent operators and non-Euclidean metrics

The power setΛ(V) of quasinilpotent operator V on a Hilbert space H is defined in [4] to study the singularity of the non-Euclidean metrics ‖(V−z)−1x‖2dz⊗dz¯ at σ(V)={0}, and it is shown that if Λ(V) contains more than one point then V has a nontrivial hyperinvariant subspace. This paper first proves that the Volterra integral operator on the classical Hardy–Hilbert space has singleton power set, thus answering a question raised in [4]. Then, it studies the length of circles under the metrics and its connection with power set. In particular, it determines the maximal length of the unit circle with respect to the change of x in the metrics. Moreover, it shows that an extremal value integral equation is able to detect representing functions for all the invariant subspaces of the Volterra operator.

Volterra type operators on [formula omitted] spaces

Čučković and Paudyal characterized the lattice of invariant subspaces of the operator T in the Hardy–Hilbert space H2(D), where they studied the special case (when p=2) of the space Sp(D). We generalize some of their works to the general case when 1≤p<∞ and determine that M is an invariant subspace of T on Hp(D) if and only if Tz(M) is an invariant subspace of Mz on S0p(D), if and only if Tz(M) is a closed ideal of S0p(D). Furthermore, we provide certain Beurling-type invariant subspaces of Mz on Sp(D) and S0p(D). Then, we investigate the boundedness of the operators Tg and Ig on Sp(D). Finally, we investigate the spectrum of multiplication operator Mg on Sp(D), the isometric multiplication operators and the isometric zero-divisors on Sp(D).

Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: A hundred decimal digits for the dimension of E2

We prove that the algorithm of [19] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set E2 (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25.The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension.

Complex symmetry of Toeplitz operators with continuous symbols

In this note, we consider the question of when a Toeplitz operator on the Hardy–Hilbert space $$H^2$$ of the open unit disk $$\mathbb {D}$$ is complex symmetric, focusing on symbols $$\phi :\mathbb {T}\rightarrow \mathbb {C}$$ that are continuous on the unit circle $$\mathbb {T}=\partial \mathbb {D}$$ . A closed curve $$\phi $$ is called nowhere winding if the winding number of $$\phi $$ is 0 about every point not in the range of $$\phi $$ . It is then shown that if $$T_\phi $$ is complex symmetric, then $$\phi $$ must be nowhere winding. Hence if $$\phi $$ is a simple closed curve, then $$T_\phi $$ cannot be a complex symmetric operator. The spectrum and invertibility of complex symmetric Toeplitz operators with continuous symbols are then described. Finally, given any continuous curve $$\gamma :[a,b]\rightarrow \mathbb {C}$$ , it is shown that there exists a complex symmetric Toeplitz operator with continuous symbol whose spectrum is precisely the range of $$\gamma $$ .

Approximation of Invariant Subspaces in Some Dirichlet-Type Spaces

The Hilbert space \(\mathcal {D}_{2}\) is the space of all holomorphic functions f defined on the open unit disc \(\mathbb {D}\) such that \({f}^{'}\) is in the Hardy Hilbert space \(\mathbf {H}^2.\) In this paper, we prove that the invariant subspaces of \(\mathcal {D}_{2}\) with respect to multiplication operator \(M_{z}\) can be approximated with finite co-dimensional invariant subspaces. We also obtain a partial result in this direction for the classical Dirichlet space.

Sequences of Weighted Composition Operators on the Hardy–Hilbert Space

Let $$\varphi $$ be an analytic self map of the open unit disc $$\mathbb {D}$$ . Assume that $$\psi $$ is an analytic map of $$\mathbb {D}$$ . Suppose that f is in the Hardy–Hilbert space of the open unit disc $$H^2$$ . The operator that takes f into $$\psi \cdot f \circ \varphi $$ is a weighted composition operator, and is denoted by $$C_{\psi ,\varphi }$$ . In this paper we relate the convergence of the sequence $$\{ C_{\psi _n,\varphi _n}\}$$ in different operator topologies to the convergence of the two sequences of maps $$\{\varphi _n \}$$ and $$\{ \psi _n\}$$ .

An embedding constant for the Hardy space of Dirichlet series

A new and simple proof of the embedding of the Hardy–Hilbert space of Dirichlet series into a conformally invariant Hardy space of the half-plane is presented, and the optimal constant of the embedding is computed.