Weighted composition operators on functional Hilbert spaces

We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreover, we show that our abstract approach unifies existing results which characterize hypercyclicity, resp. topological mixing, of (weighted) composition operators on the space of holomorphic functions on a simply connected domain in the complex plane, on the space of smooth functions on an open subset of $\mathbb{R}^d$, as well as results characterizing topological transitiviy of such operators on the space of real analytic functions on an open subset of $\mathbb{R}^d$.

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ with $$p \in (0, \infty )$$ and $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

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

We state conditions under which some classical operators acting from abstract quasi-Banach spaces of functions holomorphic in a plain domain into a weighted space of the same functions with sup-norm are compact. It is obtained abstract criteria for the compactness of a linear operator on an arbitrary quasi-Banach space which are stated in terms of delta-functions and formulate their realizations for both classical and generalized Fock spaces. The above results are applied to the weighted composition operator. It is established some conditions for the compactness of this operator which are given in terms of norms of delta-functions in the corresponding dual spaces. These results are essential generalizations of the known Zorboska’s ones. Namely, we significantly extended the class of weighted spaces of holomorphic functions with uniform norms for which one can state some conditions for the compactness of an arbitrary linear operator or the weighted composition operator.

Power bounded weighted composition operators on function spaces defined by local properties

In this work, we introduce notions of generalized firmly nonexpansive (G-firmly non expansive) and fundamentally firmly nonexpansive (F-firmly nonexpansive) mappings and utilize to the same to prove Ray's theorem for G-firmly and F-firmly nonexpansive mappings in Hilbert Spaces. Our results extend the result due to F. Kohsaka [ Ray's theorem revisited: a fixed point free firmly nonexpansive mapping in Hilbert spaces, Journal of Inequalities and Applications (2015) 2015:86 ].

In this paper, we prove a weak convergence theorem of Ishikawa's\ntype for $m$-generalized hybrid mappings in a Hilbert space. Further,\nby using a new modification of Ishikawa iteration, we\nprove a strong convergence theorem for $m$-generalized hybrid\nmappings in a Hilbert space.

We study the widely more generalized hybrid mappings which have been proposed to unify several well-known nonlinear mappings including the nonexpansive mappings, nonspreading mappings, hybrid mappings, and generalized hybrid mappings. Without the convexity assumption, we will establish the existence theorem and mean convergence theorem for attractive point of the widely more generalized hybrid mappings in a Hilbert space. Moreover, we prove a weak convergence theorem of Mann’s type and a strong convergence theorem of Shimizu and Takahashi’s type for such a wide class of nonlinear mappings in a Hilbert space. Our results can be viewed as a generalization of Kocourek, Takahashi and Yao, and Hojo and Takahashi where they studied the generalized hybrid mappings.

Iterative algorithms have been extensively studied over the class of nonexpansive mappings in Hilbert spaces. Recall that nonexpansive mappings belong to quasi-nonexpansive mappings. The aim of this article is expanding the general approximation method proposed by Marino and Xu to quasi-nonexpansive mappings in Hilbert spaces.

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.

We solve the closed range problem for weighted composition operators on Fock spaces. The result equivalently characterizes when the operators are bounded from below. We give several applications of the main result related to the operators invertibility, Fredholm, and dynamical sampling structures from frame perspectives. We prove there exists no vector in the Fock space for which its orbit under the weighted composition operator represents a frame family. Furthermore, it is shown that a weighted composition operator preserves frames if and only if it preserves the stronger Riesz basis property. Similar results are provided for the adjoint operator.

There are two aims in this paper. The first aim is to characterize the order-bounded weighted composition operators between Fock spaces, and the second is to further characterize the order-bounded difference in weighted composition operators between Fock spaces. At the same time, six examples are given to illustrate the relations between boundedness and ordered boundedness. Moreover, an interesting result is found that differences in weighted composition operators defined by some special weighted functions and symbol functions are order-bounded between Fock spaces if and only if each weighted composition operator is compact between Fock spaces. Finally, two open questions are also put forward for converting larger Fock spaces into smaller ones.

Let $\phi: \mathbb D \to \mathbb D$ and $\psi: \mathbb D \to \mathbb C$ be analytic maps. They induce a weighted composition operator $\psi C_{\phi}$ acting between weighted Banach spaces of holomorphic functions and weighted Bloch type spaces. Under s

ABSTRACTWe discover a criterion for having a closed range and completely characterize the cyclic behavior of composition operators on a class of Hilbert spaces of entire functions in the complex plane .

Optimal numerical approximation of bounded linear functionals by weighted sums in Hilbert spaces of functions analytic in a circleK r , in a circular annulusK r1,r2 and in an ellipseE r is investigated by Davis' method on the common algebraic background for diagonalising the normal equation matrix. The weights and error functional norms for optimal rules with nodes located angle-equidistant on the concentric circle∂K s or on the confocal ellipseϖE s and in the interval [−1,1] for an arbitrary bounded linear functional are given explicitly. They are expressed in terms of a complete orthonormal system in the Hilbert space.