Weighted Dirichlet Spaces Research Articles

Notes on Hypercyclicity of Composition Operator on the Weighted Dirichlet Spaces

Notes on Hypercyclicity of Composition Operator on the Weighted Dirichlet Spaces

Von Neumann’s inequality for row contractive matrix tuples

We prove that for all nin {mathbb {N}}, there exists a constant C_{n} such that for all d in {mathbb {N}}, for every row contraction T consisting of d commuting n times n matrices and every polynomial p, the following inequality holds: ‖p(T)‖≤Cnsupz∈Bd|p(z)|.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert p(T)\\Vert \\le C_{n} \\sup _{z \\in {\\mathbb {B}}_d} |p(z)| . \\end{aligned}$$\\end{document}We apply this result and the considerations involved in the proof to several open problems from the pertinent literature. First, we show that Gleason’s problem cannot be solved contractively in H^infty ({mathbb {B}}_d) for d ge 2. Second, we prove that the multiplier algebra {{,mathrm{Mult},}}({mathcal {D}}_a({mathbb {B}}_d)) of the weighted Dirichlet space {mathcal {D}}_a({mathbb {B}}_d) on the ball is not topologically subhomogeneous when d ge 2 and a in (0,d). In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra A({mathcal {D}}_a({mathbb {B}}_d)) of {{,mathrm{Mult},}}({mathcal {D}}_a({mathbb {B}}_d)) generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball mathfrak {C}mathfrak {B}_d that is levelwise uniformly continuous but not globally uniformly continuous.

Weighted composition operators on weighted Bergman and Dirichlet spaces

AbstractWe study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.

Recurrence of multiples of composition operators on weighted Dirichlet spaces

A bounded linear operator T acting on a Hilbert space $$\mathcal {H}$$ is said to be recurrent if for every non-empty open subset $$U\subset \mathcal {H}$$ there is an integer n such that $$T^n (U)\cap U\ne \emptyset$$ . In this paper, we completely characterize the recurrence of scalar multiples of composition operators, induced by linear fractional self maps of the unit disk, acting on weighted Dirichlet spaces $$\mathcal {S}_\nu$$ ; in particular on the Bergman space, the Hardy space, and the Dirichlet space. Consequently, we complete previous work of Costakis, Manoussos, and Parissis on the recurrence of linear fractional composition operators on Hardy space. In this manner, we determine the triples $$(\lambda ,\nu ,\phi )\in {\mathbb {C}}\times \mathbb {R}\times \mathrm{LFM}(\mathbb {D})$$ for which the scalar multiple of composition operator $$\lambda C_\phi$$ acting on $$\mathcal {S}_\nu$$ fails to be recurrent.

Multiplier tests and subhomogeneity of multiplier algebras

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.

Volterra Type Operators on Weighted Dirichlet Spaces

The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for the boundedness are not complete. In this paper, the author completely characterizes the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces D pα to D qβ (−1 < α, β and 0 < p < q < ∞), which essentially complete their works. Furthermore, the author investigates the order boundedness of Volterra type operators between weighted Dirichlet spaces.

Random Interpolating Sequences in Dirichlet Spaces

Abstract We discuss random interpolating sequences in weighted Dirichlet spaces ${{\mathcal{D}}}_\alpha $, $0\leq \alpha \leq 1$, when the radii of the sequence points are fixed a priori and the arguments are uniformly distributed. Although conditions for deterministic interpolation in these spaces depend on capacities, which are very hard to estimate in general, we show that random interpolation is driven by surprisingly simple distribution conditions. As a consequence, we obtain a breakpoint at $\alpha =1/2$ in the behavior of these random interpolating sequences showing more precisely that almost sure interpolating sequences for ${{\mathcal{D}}}_\alpha $ are exactly the almost sure separated sequences when $0\le \alpha &amp;lt;1/2$ (which includes the Hardy space $H^2={{\mathcal{D}}}_0$), and they are exactly the almost sure zero sequences for ${{\mathcal{D}}}_\alpha $ when $1/2 \leq \alpha \le 1$ (which includes the classical Dirichlet space ${{\mathcal{D}}}={{\mathcal{D}}}_1$).

Polynomial approximation in weighted Dirichlet spaces

We give an elementary proof of an analogue of Fejer’s theorem in weighted Dirichlet spaces with superharmonic weights. This provides a simple way of seeing that polynomials are dense in such spaces.

Cesàro Summability of Taylor Series in Weighted Dirichlet Spaces

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $$(C,\alpha )$$ -summable to the function in the norm of the space, provided that $$\alpha >1/2$$ . We further show that the constant 1/2 is sharp, in marked contrast with the classical case of the disk algebra.

Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions

We compare the rate of decay of singular numbers of a given composition operator acting on various Hilbert spaces of analytic functions on the unit disk D. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the “cusp map” and the lens maps, acting on weighted Dirichlet spaces.

Order boundedness of weighted composition operators on weighted Dirichlet spaces and derivative Hardy spaces

In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces, which generalizes the previous works done by Sharma et al.

Spectra of Composition Groups on the Weighted Dirichlet Space of the Upper Half-Plane

We prove that the group of weighted composition operators induced by continuous automorphism groups of the upper half plane $$\mathbb{U}$$ is strongly continuous on the weighted Dirichlet space of $$\mathbb{U}$$ , $${{\cal D}_\alpha }(\mathbb{U})$$ . Further, we investigate when they are isometries on $${{\cal D}_\alpha }(\mathbb{U})$$ . In each case, we determine the semigroup properties while in the case that the induced composition group is an isometry, we apply similarity theory to determine the spectral properties of the group.

On the Wandering Property in Dirichlet spaces

We show that in a scale of weighted Dirichlet spaces $$D_{\alpha }$$, including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in $$D_{\alpha }$$ such that B satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell et al. (Indiana Univ Math J 51(4):931–961, 2002). As a particular instance, when $$B(z)=z^k$$ and $$|\alpha | \le \frac{\log (2)}{\log (k+1)}$$, the chosen norm is the usual one in $$D_\alpha $$.

Multipliers for Dirichlet type spaces by randomization

Cochran et al. (Can J Math 45(2):255–268, 1993) proved that if $$f(z)=\sum _{n=0}^{\infty }a_nz^n$$ is a holomorphic function in the Dirichlet space $${\mathcal {D}}$$ of analytic functions in the unit disc, then for almost every choice of signs the function given by $$\sum _{n=0}^{\infty }\pm a_nz^n$$ is a multiplier of $${\mathcal {D}}.$$ They obtain also versions of this result for the weighted Dirichlet spaces $$D_\alpha ^2 \ (0\le \alpha <1).$$ The purpose of this paper is to generalized these results to the setting of the spaces of Dirichlet type $$D_\alpha ^p \ (0 -1).$$ We prove that, if $$1\le p<\infty $$ and $$-1<\alpha <p-1,$$ then $$\sum _{n=0}^{\infty }\pm a_nz^n$$ is a multiplier of $$D_\alpha ^p$$ almost surely if and only if $$\sum _{n=0}^{\infty }2^{n(p-\alpha -1)}\Big (\sum _{k=2^n}^{2^{n+1}-1}|a_k|^2\Big )^{p/2}<\infty .$$ Also, we give some sufficient conditions for the two cases where $$\alpha >-1$$ and $$0<p\le \alpha +1,$$ and $$\alpha >-1$$ and $$\alpha +1\le p\le 1.$$ Some related issues for lacunary series are discussed too.

Blaschke Products and Zero Sets in Weighted Dirichlet Spaces

In this paper, we deal with superharmonically weighted Dirichlet spaces $\mathcal {D}_{\omega }$ . First, we prove that the classical Dirichlet space is the largest, among all these spaces, which contains no infinite Blaschke product. Next, we give new sufficient conditions on a Blaschke sequence to be a zero set for $\mathcal {D}_{\omega }$ . Our conditions improve Shapiro-Shields condition for $\mathcal {D}_{\alpha }$ , when α ∈ (0,1).

Hadamard Multipliers on Weighted Dirichlet Spaces

The Hadamard product of two power series is obtained by multiplying them coefficientwise. In this paper we characterize those power series that act as Hadamard multipliers on all weighted Dirichlet spaces on the disk with superharmonic weights, and we obtain sharp estimates on the corresponding multiplier norms. Applications include an analogue of Fej\'er's theorem in these spaces, and a new estimate for the weighted Dirichlet integrals of dilates.

Some Characterizations of Composition Operators on Weighted Dirichlet Spaces

In this paper, we give three different characterizations for the boundedness and compactness of composition operators between different weighted Dirichlet spaces in the unit disk.

Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-trees

In this note we investigate the multi-parameter Potential Theory on the weighted d-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish some basic properties of the respective potentials, capacities and equilibrium measures (in particular in the case of product polynomial weights). We explore multi-parameter Hardy inequality and its trace measures, and discuss some open problems of potential-theoretic and combinatorial nature.

Power bounded weighted composition operators and power bounded below composition operators

In this paper, we characterize power bounded weighted composition operators on weighted Bergman spaces of strongly pseudoconvex bounded domains in $${\mathbb {C}}^n$$. Also, we introduce the notion of power bounded below operators, then, for $$\alpha >0$$, we characterize power bounded below composition operators on $${\mathcal {D}}_\alpha $$, the weighted Dirichlet space on the unit disk of the complex plane.

Paley–Wiener Theorems on the Siegel Upper Half-Space

In this paper we study spaces of holomorphic functions on the Siegel upper half-space $\mathcal U$ and prove Paley-Wiener type theorems for such spaces. The boundary of $\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. Using the group Fourier transform on $\mathbb H_n$, Ogden-Vagi proved a Paley-Wiener theorem for the Hardy space $H^2(\mathcal U)$. We consider a scale of Hilbert spaces on $\mathcal U$ that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury-Arveson space, and the Dirichlet space $\mathcal D$. For each of these spaces, we prove a Paley-Wiener theorem, some structure theorems, and provide some applications. In particular we prove that the norm of the Dirichlet space modulo constants $\dot{\mathcal D}$ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of $\mathcal U$.