Weighted Fock Spaces Research Articles

Uncertainty inequality for weighted Fock spaces

Uncertainty inequality for weighted Fock spaces

Composition operators on weighted Fock spaces induced by $$A_{\infty }$$-type weights

Composition operators on weighted Fock spaces induced by $$A_{\infty }$$-type weights

Toeplitz operators on non-reflexive Fock spaces

We generalize several results on Toeplitz operators over reflexive, standard weighted Fock spaces F_t^p to the non-reflexive cases p = 1, \infty . Among these results are the characterization of compactness and the Fredholm property of such operators, a well-known representation of the Toeplitz algebra, and a characterization of the essential center of the Toeplitz algebra. Further, we improve several results related to correspondence theory, e.g., we improve previous results on the correspondence of algebras and we give a correspondence theoretic version of the well-known Berger–Coburn estimates.

Generalized composition operators on weighted Fock spaces

The generalized composition operators $J_g^{\varPhi}$ and $C_g^{\varPhi}$, induced by analytic functions $g$ and $\varPhi$ on the complex plane $\mathbb{C}$, are defined by \begin{align*} J_g^{\varPhi}(f)(z)=\int\limits_0^zf'(\varPhi(\omega))g(\omega)d\omega and C_g^{\varPhi}(f)(z)=\int\limits_0^{\varPhi(z)}f'(\omega)g(\omega)d\omega. \end{align*} In this paper, we consider these operators on weighted Fock spaces $\mathcal{F}_p^{\varPsi}$, consisting of entire functions, which are $\mathcal{L}^p(\mathbb{C})$-integrable with respect to the measure $d\lambda(z)=e^{-\varPsi (z)}d\Lambda(z)$, where $d\Lambda$ is the usual Lebesgue area measure in $\mathbb{C}$. We assume that the weight function $\varPsi$ in the spaces satisfies certain smoothness conditions, in particular, this weight function grows faster than the Gaussian weight $\frac{|z|^2}{2}$ defining the classical Fock spaces. We first consider bounded and compact properties of $J_{g}^ {\varPhi}$ and $C_{g}^{\varPhi}$, and characterize these properties in terms function theory of inducing functions $g$ and $\varPhi$, given by \begin{align*} \mathcal{M}_g^{\varPhi}(z):=\frac{|g(z)|\varPsi'(\varPhi(z))}{1+\varPsi'(z)}e^{\varPsi(\varPhi(z))-\varPsi(z)}. \end{align*} Our characterization is simpler to use than the Berezin type integral transform characterization. In some cases, our result shows that these operators experience poorer boundedness and compactness structures when acting between such spaces than the classical Fock spaces. For instance, for $\varPhi(z)=z$, there is no nontrivial bounded $J_g^{\varPhi}$ and $C_g^{\varPhi}$ on weighted Fock spaces. In the case of classical Fock spaces, they are bounded if and only if $g$ is constant. In the second part of this paper, we apply our simpler characterization of boundedness and compactness to further study the Schatten-class membership of these operators. In particular, we express the Schatten $S_p(\mathcal{F}_2^{\varPsi})$ class membership property in terms of $\mathcal{L}^p(\mathbb{C}, \Delta \varPsi d\Lambda)$-integrability of $\mathcal{M}_g^{\varPhi}$.

A Note on the Differential Operator on Weighted Fock Spaces

The differential operator Df = f′ is a canonical example of unbounded operators in many Banach spaces. In this study, we identify a class of weighted Fock spaces in which the differential operator is bounded. Then, we obtain some properties of the differential operator on these spaces in terms of its C*-algebraic structure in the one and two-dimensional cases.

Small Hankel operators on generalized weighted Fock spaces

In this work we characterize the boundedness, compactness and membership in the Schatten class of small Hankel operators on generalized weighted Fock spaces F α p , ℓ ( ω ) F^{p,\ell }_\alpha (\omega ) associated to an A p ℓ \mathcal {A}^\ell _p weight ω \omega , for 1 > p > ∞ 1>p>\infty , ℓ ≥ 1 \ell \ge 1 , and α > 0 \alpha >0 .

Noncommutative Varieties, Symmetric Weighted Fock Spaces, and Multipliers

Noncommutative Varieties, Symmetric Weighted Fock Spaces, and Multipliers

Factorization of the Fock–Sobolev space

It is well known that polynomially growing/decaying weights come into play in the study of Fock–Sobolev spaces. Given an integer m we introduce the weighted Fock spaces F m p that contain all entire functions f such that ( 1 + | z | ) m f ( z ) ∈ F p . For a positive integer m we have P m ⋅ F m p = F p and P m ⋅ F p = F − m p , where P m is the ring of all polynomials q ( z ) on C satisfying deg ⁡ ( q ) ≤ m .

A new explicit complete Kähler metric on ℂ n

In 1977, Klembeck presented an explicit Kähler metric of positive holomorphic curvature on [Klembeck P. A complete Kähler metric of positive curvature on . Proc Amer Math Soc. 1977;64(2):313–316]. Later, in 1991, To used Klembeck's metric and proved a non-compact version of the Kodaira embedding theorem [To W-K. Quasi-projective embeddings of noncompact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions. Duke Math J. 1991;63(3):745–789]. Wu H-H and Zheng F. [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] discovered another explicit Kähler metric which is a natural perturbation of Klembeck's example. More examples (implicit) are studied by solutions of certain ODEs by Cao H-D. [Existence of gradient Kähler-Ricci solitons. In: Elliptic and parabolic methods in geometry. Minnesota: 1994. p. 1–16; Limits of solutions to the Kähler-Ricci flow. J Differ Geom. 1997;45:257–272] and Wu-Zheng [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] respectively. Motivated by these, we investigate other complete Kähler metrics induced from certain weighted Fock spaces, following a similar scheme. In particular, we prove that these Bergman metrics induced by weighted Fock spaces possess positive holomorphic sectional curvature.

Random zero sets for Fock type spaces

Given a nondecreasing sequence \(\Lambda =\{\lambda _n>0\}\) such that \(\displaystyle \lim _{n\rightarrow \infty } \lambda _n=\infty ,\) we consider the sequence \(\mathcal {N}_\Lambda :=\left\{ \lambda _ne^{i\theta _n},n\in \,\mathbb {N}\right\} \), where \(\theta _n\) are independent random variables uniformly distributed on \([0,2\pi ].\) We discuss the conditions on the sequence \(\Lambda \) under which \(\mathcal N_\Lambda \) is a zero set (a uniqness set) of a given weighted Fock space almost surely. The critical density of the sequence \(\Lambda \) with respect to the weight is found.

Hilbert-Type Operators Acting Between Weighted Fock Spaces

In this paper we introduce and study several new Hilbert-type operators acting between the weighted Fock spaces. We provide some sufficient and necessary conditions for the boundedness and compactness of certain Hilbert-type operators from one weighted Fock space to another.

Korenblum constants for various weighted Fock spaces

We study the Korenblum Maximum Principle for various weighted Fock spaces. The main tool relies on applying special cases of Ramanujan's Master Theorem involving the Gamma function and the Mellin transform of the Dirichlet series. It is interesting that with elementary probe functions, we still can obtain closed-form upper bound expressions, in terms of several well-known special functions, for the Korenblum constants of various weighted Fock spaces. For the first time, we obtain upper bounds of the Korenblum constant for the finite and infinite intersections of weighted Fock spaces and proved that the Korenblum Maximum Principle fails for those infinite intersections of weighted Fock spaces. Some open questions are provided.

Fredholm Toeplitz Operators on Doubling Fock Spaces

Recently the authors characterized the Fredholmn properties of Toeplitz operators on weighted Fock spaces when the Laplacian of the weight function is bounded below and above. In the present work the authors extend their characterization to doubling Fock spaces with a subharmonic weight whose Laplacian is a doubling measure. The geometry induced by the Bergman metric for doubling Fock spaces is much more complicated than that of the Euclidean metric used in all the previous cases to study Fredholmness, which leads to considerably more involved calculations.

Composition operators and Gleason’s problem on weighted Fock spaces

Composition operators and Gleason’s problem on weighted Fock spaces

Compact Hankel operators with bounded symbols

We give a new proof of the result that the Hankel operator Hf with a bounded symbol is compact on standard weighted Fock spaces F2α(Cn) if and only if H¯¯¯f is compact. Our proof uses limit operator techniques and extends to Fpα(Cn) when $1

On the dimension of the Fock type spaces

We study the weighted Fock spaces in one and several complex variables. We evaluate the dimension of these spaces in terms of the weight function extending and completing earlier results by Rozenblum–Shirokov and Shigekawa.

Multiple Sampling and Interpolation in Weighted Fock Spaces of Entire Functions

We characterize sampling and interpolating sets with derivatives in weighted Fock spaces on the complex plane in terms of their weighted Beurling densities.

Explicit formula for the reproducing kernels for some weighted Fock spaces

We introduce new weighted Fock spaces Fˆ−α2 identified to weighted Fock spaces F−α2 with equivalent norms. Then the study of the weighted Fock spaces F−α2 changes to that of the weighted Fock spaces Fˆ−α2. Moreover we can compute reproducing kernels for Fˆ−α2, explicitly.

Composition Operators Between Weighted Fock Spaces

In this paper, we characterize completely the boundedness and compactness of composition operators between weighted Fock spaces.

Heat and Laplace type equations with complex spatial variables in weighted Fock spaces

In a recent book co-authored by the authors of this article, we studied by semigroup theory methods several classical evolution equations, including the heat and Laplace equations, with real time variable and complex spatial variable, under the hypothesis that the boundary function belongs to the space of analytic functions in the open unit disk and continuous in the closed unit disk, endowed with the uniform norm. Also, in a subsequent paper, the authors have extended the results for the heat and Laplace equations in weighted Bergman spaces on the unit disk. The purpose of this article is to show that the semigroup theory methods work for these two evolution equations of complex spatial variables, under the hypothesis that the boundary function belongs to the weighted Fock space on \(\mathbb{C}\), \(F_{\alpha }^p(\mathbb{C})\), with \(1\leq p<+\infty \), endowed with The \(L^p\)-norm. Also, the case of several complex variables is considered. The proofs use the Jensen's inequality, Fubini's theorem for integrals and the \(L^p\)-integral modulus of continuity.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/109/abstr.html