Volterra Type Operators on Weighted Dirichlet Spaces

We study the composition operators with closed range on the weighted Bloch spaces, the weighted Dirichlet spaces, the Bergman spaces and the Hardy space, the space BMOA. In particular, we study the relationship between with closed range on the weighted Bloch spaces and with closed range on the weighted Dirichlet spaces.

Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S 2, the space of analytic functions whose first derivative is in H 2, and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.

In this paper, linear combinations of composition operators acting on weighted Dirichlet spaces are studied. By using the first derivative of the kernel function, we obtain a lower estimate for the essential norms of these operators acting on the Dirichlet space D and S2. For general weighted Dirichlet space, by using complex interpolation methods, we characterize the compactness of these operators induced by linear fractional self-maps of the disk.

We characterize bounded multiplication operators in weighted Dirichlet spaces that are power bounded, Cesàro bounded and uniformly Kreiss. Moreover, we show the equivalence in such spaces between mean ergodicity and Cesàro boundedness for multiplication operators. We perform the same study for adjoints of multiplication operators. As a particular example, we obtain a uniform mean ergodic multiplication operator in Dirichlet spaces that fails to be power bounded.

Normal weighted composition operators on weighted Dirichlet spaces

We characterize bounded and compact composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function for the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. We also present several examples and counter-examples that point out the borderlines of the result and its connections to other themes.

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.

Cantor sets and cyclicity in weighted Dirichlet spaces

We investigate isometric composition operators on the weighted Dirichlet space \({D_\alpha }\) with standard weights \({(1 - {\left| z \right|^2})^\alpha },\alpha > - 1\). The main technique used comes from Martin and Vukotic who completely characterized the isometric composition operators on the classical Dirichlet space D. We solve some of these but not in general. We also investigate the situation when \({D_\alpha }\) is equipped with another equivalent norm.

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.

In this note, using some conditions on the weight function K, we investigate the inner functions as multipliers in weighted Dirichlet spaces, and we also discuss zero sets.

We study composition operators acting on weighted Dirichlet spaces. We obtain estimates for the essential norm, describe the membership in Schatten–Von Neumann ideals and characterize the composition operators with closed range.

We consider holomorphic functions $\phi$ taking the unit disc $U$ into itself, and Banach spaces $X$ consisting of functions holomorphic in $U$ and continuous on its closure; and show that under some natural hypotheses on $X$: if $\phi$ induces a compact composition operator on $X$, then $\phi (U)$ must be a relatively compact subset of $U$. Spaces $X$ which satisfy the hypotheses of this theorem include the disc algebra, "heavily" weighted Dirichlet spaces, spaces of holomorphic Lipschitz functions, and the space of functions with derivative in a Hardy space ${H^p}(p \geq 1)$. It is well known that the theorem is not true for "large" spaces such as the Hardy and Bergman spaces. Surprisingly, it also fails in "very small spaces," such as the Hilbert space of holomorphic functions $f(z) = \sum {{a_n}{z^n}}$ determined by the condition $\sum {|{a_n}{|^2}\exp \left ( {\sqrt n } \right ) < \infty }$. The property of Möbiusinvariance plays a crucial and mysterious role in these matters.

In this paper, we mainly study the frequent hypercyclicity of a scalar multiple of the tensor product of two linear fractional composition operators on , where is the weighted Dirichlet space. We will restrict the discussion to the situations that both φ and ψ are hyperbolic non automorphisms, hyperbolic automorphisms and parabolic automorphisms, respectively. Meanwhile, we give the equivalent conditions of to be frequently hypercyclic, hypercyclic, mixing and chaotic.

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.