arXiv subjects
Publications and source records attributed to Tesfa Mengestie.
We characterize all pairs of entire functions $(u,ψ)$ for which the induced weighted superposition operator $S_{(u,ψ)}$ transforms one Fock space into another Fock space.Further analytical structures like boundedness and Lipschitz continuity of $S_{(u,ψ)}$ are described. We, in particular, show the Fock spaces support no compact weighted superposition operator.
We identify Fock-type spaces $\mathcal{F}_{(m,p)}$ on which the differentiation operator $D$ has closed range. We prove that $D$ has closed range only if it is surjective, and this happens if and only if $m=1$. Moreover, since the operator is unbounded on the classical Fock spaces, we consider the modified or the weighted composition--differentiation operator, $D_{(u,ψ,n)} f= u\cdot\big( f^{(n)}\circ ψ\big)$, on these spaces and describe conditions under which the operator admits closed range, surjective, and order bounded structures.
We characterize the convex-cyclic weighted composition operators $W_{(u,ψ)}$ and their adjoints on the Fock space in terms of the derivative powers of $ ψ$ and the location of the eigenvalues of the operators on the complex plane. Such a description is also equivalent to identifying the operators or their adjoints for which their invariant closed convex sets are all invariant subspaces. We further show that the space supports no supercyclic weighted composition operators with respect to the pointwise convergence topology and hence with the weak and strong topologies and answers a question raised by T. Carrol and C. Gilmore in \cite{CC}.
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.
For pairs of holomorphic maps $(u,ψ)$ on the complex plane, we study some dynamical properties of the weighted composition operator $W_{(u,ψ)}$ on the Fock spaces. We prove that no weighted composition operator on the Fock spaces is supercyclic. Conditions under which the operators satisfy the Ritt's resolvent growth condition are also identified. In particular, we show that a non-trivial composition operator on the Fock spaces satisfies such a growth condition if and only if it is compact.
For holomorphic pairs of symbols $(u, ψ)$, we study various structures of the weighted composition operator $ W_{(u,ψ)} f= u \cdot f(ψ)$ defined on the Fock spaces $\mathcal{F}_p$. We have identified operators $W_{(u,ψ)}$ that have power bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values $|u(0)|$ and $|u(\frac{b}{1-a})|$ where $ a$ and $b$ are coefficients from linear expansion of the symbol $ψ$. The spectrum of the operators are also determined and applied further to prove results about uniform mean ergodicity.
We study various properties of composition operators acting between generalized Fock spaces $\mathcal{F}_φ^p$ and $\mathcal{F}_φ^q$ with weight functions $φ$ grow faster than the classical Gaussian weight function $\frac{1}{2}|z|^2$ and satisfy some mild smoothness conditions. We have shown that if $p\neq q,$ then the composition operator $C_ψ: \mathcal{F}_φ^p \to \mathcal{F}_φ^q $ is bounded if and only if it is compact. This result shows a significance difference with the analogous result for the case when $C_ψ$ acts between the classical Fock spaces or generalized Fock spaces where the weight functions grow slower than the Gaussian weight function. We further described the Schatten $\mathcal{S}_p(\mathcal{F}_φ^2)$ class, normal, unitary, cyclic and supercyclic composition operators. As an application, we characterized the compact differences, the isolated and essentially isolated points, and connected components of the space of the operators under the operator norm topology.
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.
This paper is a follow-up contribution to our work [10] where we studied some spectral properties of the differential operator $D$ acting between generalized Fock spaces $\mathcal{F}_{(m,p)}$ and $\mathcal{F}_{(m,q)}$ when both exponents $p$ and $q$ are finite. In this note we continue to study the properties for the case when at least one of the spaces is growth type. We also estimate the essential norm of $D: \mathcal{F}_{(m,p)}\to \mathcal{F}_{(m,q)}$ for all $1\leq p, q\leq \infty$, and showed that if the operator fails to be compact, then its essential norm is comparable to the operator norm and $\|D\|_e \simeq \big|m^{2+p}-m^{1+p}\big|^{\frac{1}{p}}\simeq \|D\|.$
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.
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.
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.
It has long been known that the differential operator $D$ represents a typical examples of unbounded operators in many Banach spaces including the classical Fock spaces, the Fock--Sobolev spaces, and the generalized Fock spaces where the weight decays faster than the Gaussian weight. In this note we identify Fock type spaces where the operator admits some basic spectral structures including compactness and membership in the Schatten $\mathcal{S}_p$ classes. We also showed that its nontrivial spectrum while acting on such spaces is precisely the closed unit disk $\D$ in the complex plane.
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 coefficient of the highest degree term in a polynomial expansion of $g$.
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.
We study some mapping properties of Volterra type integral operators and composition operators on model spaces. We also discuss and give out a couple of interesting open problems in model spaces where any possible solution of the problems can be used to study a number of other operator theoretic related problems in the spaces.
We study the Schatten class membership of generalized Volterra companion integral operators on the standard Fock spaces $\mathcal{F}_α^2$. The Schatten $\mathcal{S}_p(\mathcal{F}_α^2)$ membership of the operators are characterized in terms of $L^{p/2}$ integrability of certain generalized Berezin type integral transforms on the complex plane. We also give a more simplified and easy to apply description in terms of $L^p$ integrability of the symbols inducing the operators against a supper exponentially decreasing weights. An asymptotic estimates for the $\mathcal{S}_p(\mathcal{F}_α^2)$ norms of the operators have been also provided.
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.