Weighted composition operators on weighted Bergman spaces and weighted Bloch 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 characterize bounded and compact weighted composition operators acting between weighted Bergman spaces and between Hardy spaces. Our results use certain integral transforms that generalize the Berezin transform. We also estimate the essential norms of these operators. As applications, we characterize bounded and compact pointwise multiplication operators between weighted Bergman spaces and estimate their essential norms.

We study when a weighted composition operator acting between different weighted Bergman spaces is bounded, resp. compact.

We study the boundedness of weighted composition operators acting between weighted Bergman spaces.

Let $\varphi_{1}$ and $\varphi_{2}$ be holomorphic self-maps of the open unit ball $\mathbb B$ in $\mathbb C^N$, $u_{1}$ and $u_{2}$ be holomorphic functions on $\mathbb B$ and let weighted composition operators $W_{\varphi_{1},u_{1}}$; $W_{\varphi_{2},u_{2}}: A^{p}_{\alpha} \to H^{\infty}_{v}$ be bounded. This paper characterizes the compactness of the difference of these operators from the weighted Bergman space $A^p_\alpha$, $0-1$, to the weighted-type space $H^\infty_v$ of holomorphic functions on $\mathbb B$ in terms of inducing symbols $\varphi_{1}$, $\varphi_{2}$, $u_{1}$ and $u_{2}$. For the case $p \gt 1$ we find an asymptotically equivalent expression to the essential norm of the operator.

Let σ be a Békollé weight function and ν be a weight function. In this paper, we characterize the boundedness and compactness of weighted composition operators acting from Bergman-type spaces $A^{p}(\sigma)$ to Bloch-type spaces $\mathcal{B}_{\nu}$ and $\mathcal{B}_{\nu, 0}$ , considerably extending some results in the literature.

We study the weighted composition operators Wh,φ on Hardy space \(H^2 (\mathbb{B})\) whenever h ∈ BMOA (resp. h ∈ VMOA). Analogous results are given for \(H^p (\mathbb{B})\) spaces and the scale of weighted Bergman spaces. In the latter case, BMOA is replaced by the Bloch space (resp. VMOA by the little Bloch space).

Let $\varphi,\psi$ be the analytic self-maps of the unit ball ${\mathbb B}$, we characterize the Hilbert-Schmidt differences of two composition operator $C_\varphi$ and $C_\psi$ on weighted Bergman space $A_\alpha^2$, and give some conclusions about the topological structure of $\mathcal{C}(A_\alpha^2)$, the space of all bounded composition operators on $A_\alpha^2$ endowed with operator norm.

In this paper we study bounded and compact weighted composition operator, induced by a fixed analytic function and an analytic self-map of the open unit disk, from Bergman space into weighted Bloch space. As a corollary, obtain the characterization of composition operator from Bergman space into weighted Bloch space.

Isometric weighted composition operators on weighted Bergman spaces

Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball

In this paper, we investigate when weighted composition operators acting on Dirichlet spaces [Formula: see text] are complex symmetric with respect to some special conjugations. We then provide some characterizations of Hermitian weighted composition operators on [Formula: see text]. Furthermore, we give a sufficient and necessary condition for [Formula: see text]-symmetric weighted composition operators on Hardy spaces [Formula: see text] to be unitary or Hermitian, then some new examples of complex symmetric weighted composition operators on [Formula: see text] are obtained. We also discuss the normality of complex symmetric weighted composition operators on [Formula: see text].

Let A p α (B n ; C d ) be the weighted Bergman space on the unit ball B n of C n of functions taking values in C d .For 1 < p < ∞ let T p,α be the algebra generated by finite sums of finite products of Toeplitz operators with bounded matrix-valued symbols (this is called the Toeplitz algebra in the case d = 1).We show that every S ∈ T p,α can be approximated by localized operators.This will be used to obtain several equivalent expressions for the essential norm of operators in T p,α .We then use this to characterize compact operators in A p α (B n ; C d ).The main result generalizes previous results and states that an operator in A p α (B n ; C d ) is compact if only if it is in T p,α and its Berezin transform vanishes on the boundary.

&lt;abstract&gt;&lt;p&gt;The main goal of this paper was to completely characterize complex symmetric difference of the weighted composition operators induced by three type symbols on weighted Bergman space of the right half-plane with the conjugations $ \mathcal{J}f(z) = \overline{f(\bar{z})} $, $ \mathcal{J}_sf(z) = \overline{f(\bar{z}+is)} $, and $ \mathcal{J}_*f(z) = \frac{1}{z^{{\alpha}+2}}\overline{f(\frac{1}{\bar{z}})} $. The special phenomenon that we focus on is that the difference is complex symmetric on weighted Bergman spaces of the half-plane with the related conjugation if and only if each weighted composition operator is complex symmetric.&lt;/p&gt;&lt;/abstract&gt;

Let be the unit ball in the -dimensional complex space, for , a holomorphic function in , and , a holomorphic map from into itself, the weighted composition operator on the weighted Hardy space is given by , where . This paper discusses the spectrum of when it is compact on a certain class of weighted Hardy spaces and when the composition map has only one fixed point inside the unit ball.