Volterra companion integral and multiplication operators with holomorphic symbols are studied for a large class of generalized Fock spaces on the complex plane $\CC$. The weights defining these spaces are radial and subject to a mild smoothness condition. In addition, we assumed that the weights decay faster than the classical Gaussian weight. One of our main results show that there exists no nontrivial holomorphic symbols $g$ which induce bounded Volterra companion integral $I_g$ and multiplication operators $M_g$ acting between the weighted spaces. We also describe the bounded and compact Volterra-type integral operators $V_g$ acting between $\mathcal{F}_q^ψ$ and $\mathcal{F}_p^ψ$ when at least one of the exponents $p$ or $q$ is infinite, and extend results of Constantin and Peláez for finite exponent cases. Furthermore, we showed that the differential operator $D$ acts in unbounded fashion on these and the classical Fock spaces.
In this paper, we obtain an isometry between the Fock-Sobolev space and the Gauss-Sobolev space. As an application, we use multipliers on the Gauss-Sobolev space to characterize the boundedness of an integral operator on the Fock-Sobolev space.
We characterize the bounded and compact generalized Volterra companion integral operators on Fock spaces acting between the standard Fock spaces. As a special case, we prove that there exist no nontrivial compact Volterra companion integral and multiplication operators on Fock spaces. We also obtain asymptotic estimates for the norm of these operators.
We study the cyclic and supercyclic dynamical properties of weighted composition operators on the Fock space $\mathcal{F}_2$. A complete characterization of cyclicity which depends on the derivative of the symbol for the composition operator and zeros of the weight function is provided. It is further shown that the space fails to support supercyclic weighted composition operators. As a consequence, we also noticed that the space supports no cyclic multiplication operator.
In this paper we introduce and study a two-parameter family of integral operators on the Fock space $F^2(C)$. We determine exactly when these operators are bounded and when they are unitary. We show that, under the Bargmann transform, these operators include the classical linear canonical transforms as special cases. As an application, we obtain a new unitary projective representation for the special linear group $SL(2,R)$ on the Fock space.
We determine the spectrum of the Voltterra-type integral operators $V_g$ on the growth type Fock--Sobolev spaces $\mathcal{F}_{ψ_m}^\infty$. We also characterized the bounded and compact spectral properties of the operators in terms of function-theoretic properties of the inducing map $g$. As a means to prove our main results, we first described the spaces in terms of Littlewood--Paley type formula which is interest of its own.
Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt's resolvent condition if and only if it is power bounded and uniformly mean ergodic. Some similar results are obtained for the Volterra-type and Hardy integral operators.
Let $μ$ be a positive Borel measure on the positive real axis. We study the integral operator $$ \mathcal{H}_μ(f)(z)=\int_{0}^{\infty}\frac{1}{t}f\left(\frac{z}{t}\right)\,dμ(t),\quad z\in \mathbb{C}\,, $$ acting on the Fock spaces $F^{p}_α$, $p\in [1,\infty],\,α>0$. Its action is easily seen to be a coefficient multiplication by the moment sequence $$ μ_n= \int_{1}^{\infty}\frac{1}{t^{n+1}}\,dμ(t) . $$ We prove that \begin{equation*} ||\mathcal{H}_μ||_{F^{p}_α\to F^{p}_α}=\sup_{n\in\mathbb{N}}μ_n,\,\,\,\,\,1\leq p\leq \infty\,\,. \end{equation*} A little-o,condition describes the compactness …
In this paper, a new two-dimensional Hardy type inequality is given in terms of pseudo-analysis dealing with set-valued functions. The first one is given for a pseudo-integral of set-valued function where pseudo-addition and pseudo-multiplication are constructed by a monotone continuous function $g:[0, \infty ]\to[0, \infty]$. Another is given by the semiring $([0, 1], \max, \odot)$ where pseudo-multiplication is generated by an increasing continuous function.
Integral operators, embedding theorems and a Littlewood-Paley formula on weighted Fock spaces
我们得到一个完整的整个功能$g$的特征，使积分算子$(T_ g f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ$是有界的或紧的，在一大类的Fock空间$\mathcal{F}^ϕ_p$，诱导光滑的径向权重比经典的高斯衰减快。在某些方面，这些空间与经典的Fock空间有很大的不同。还提供了Schatten类积分算子的描述。证明了$||\cdot||_{\mathcal{F}^ϕ_p}$的一个Littlewood-Paley公式，并刻画了正Borel测度，其中$\mathcal{F}^ϕ_p\subset L^q(μ)$，$0<p,q<\infty$ .此外，我们还解决了描述的子空间的$\mathcal{F}^ϕ_p$是经典的Volterra积分算子下不变的问题。
Bounded and compact product of Volterra type integral and composition operators acting between weighted Fock spaces are described. We also estimate the norms of these operators in terms of Berezin type integral transforms on the complex plan $\mathbb{C}$. All our results are valid for weighted composition operators acting on the class of Fock spaces considered under appropriate interpretation of the weights.
We study some spectral properties of Volterra-type integral operators $V_g$ and $I_g$ with holomorphic symbol $g$ on the Fock--Sobolev spaces $\mathcal{F}_{ψ_m}^p$. We showed that $V_g$ is bounded on $\mathcal{F}_{ψ_m}^p$ if and only if $g$ is a complex polynomial of degree not exceeding two, while compactness of $V_g$ is described by degree of $g$ being not bigger than one. We also identified all those positive numbers $p$ for which the operator $V_g$ belongs to the Schatten $\mathcal{S}_p$ classes. Finally, we characterize the spectrum of $V_g$ in terms of a closed disk of radius twice the c…
The Bargmann-Fock-Segal space plays an important role in mathematical physics, and has been extended into a number of directions. In the present paper we imbed this space into a Gelfand triple. The spaces forming the Fréchet part (i.e. the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and G. Salomon.
We study the topological structure of the space of Volterra-type integral operators on Fock spaces endowed with the operator norm. We proved that the space has the same connected and path connected components which is the set of all compact operators acting on the Fock spaces. We also obtained a characterization of isolated points of the space of the operators and showed that there exists no essentially isolated Volterra-type integral operator.
The compactness of the commutators of bilinear fractional integral operators and point-wise multiplication, acting on products of Lebesgue spaces, is characterized in terms of appropriate mean oscillation properties of their symbols. The compactness of the commutators when acting on product of weighted Lebesgue spaces is also studied.
We describe the $(p,q)$ Fock--Carleson measures for weighted Fock--Sobolev spaces in terms of the objects $(s,t)$-Berezin transforms, averaging functions, and averaging sequences on the complex space $\mathbb{C}^n$. The main results show that while these objects may have growth not faster than polynomials to induce the $(p,q)$ measures for $q\geq p$, they should be of $L^{p/(p-q)}$ integrable against a weight of polynomial growth for $q<p$. As an application, we characterize the bounded and compact weighted composition operators on the Fock--Sobolev spaces in terms of certain Berezin type inte…
A Characterization of Complex Symmetric Weighted Composition Operators on the Fock Space
School of Mathematics, Southwestern University of Finance and Economics（西南财经大学数学学院）；School of Mathematics and Computer Science, Shanxi Normal University（山西师范大学数学与计算机科学学院）
We show that for an entire function $φ$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator \begin{eqnarray*} S_φF(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w})\,dλ(w), \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ φ(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ Here $dλ(w)= π^{-n}e^{-\left…
We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of different…
In this note we define and study a Hilbert space-valued stochastic integral of operator-valued functions with respect to Hilbert space-valued measures. We show that this integral generalizes the classical Ito stochastic integral of adapted processes with respect to normal martingales and the Ito integral in a Fock space
We study the generalized Volterra-type integral and composition operators acting on the classical Fock spaces. We first characterize various properties of the operators in terms of growth and integrability conditions which are simpler to apply than those already known Berezin type characterizations. Then, we apply these conditions to study the compact and Schatten $\mathcal{S}_p$ class difference topological structures of the space of the operators. In particular, we proved that the difference of two Volterra-type integral operators is compact if and only if both are compact.
In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying…
