Numerical range of Toeplitz and weighted composition operators on weighted Bergman 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.

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are \({(1 - \overline{w}z)^{-\kappa}}\) for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.

We study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.

Weighted composition operators between Hilbert spaces of analytic functions in the operator norm and Hilbert–Schmidt norm topologies

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.

Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of [formula omitted

In this paper, we characterize the bonudedness and compactness of weighted composition operators from weighted Bergman spaces to weighted Bloch spaces. Also, we investigate weighted composition operators on weighted Bergman spaces and extend the obtained results in the unit ball of $mathbb{C}^n$.

Let $\varphi $ be an analytic self-map of the open unit disk $\mathbb {D}$ and let $\psi $ be an analytic function on $\mathbb {D}$ such that the weighted composition operator $C_{\psi ,\varphi }$ defined by $C_{\psi ,\varphi }(f)=\psi f\circ \varphi $ is bounded on the Hardy and weighted Bergman spaces. We characterize those weighted composition operators $C_{\psi ,\varphi }$ on $H^{2}$ and $A_{\alpha }^{2}$ that are essentially hypo-normal, when $\varphi $ is a linear-fractional non-automorphism.

In this paper, we study weighted composition operators and differences of composition operators. We use the Carleson measure to estimate the operator norm and the essential norm of those operators which act between two Bergman spaces with Békollé weights.

In this paper we will show how the boundedness condition for the weighted composition operators on a class of spaces of analytic functions on the open right complex half-plane called Zen spaces (which include the Hardy spaces and weighted Bergman spaces) can be stated in terms of Carleson measures and Bergman kernels. In Hilbertian setting we will also show how the norms of causal weighted composition operators on these spaces are related to each other and use it to show that an (unweighted) composition operatorC_varphi is bounded on a Zen space if and only if varphi has a finite angular derivative at infinity. Finally, we will show that there is no compact composition operator on Zen spaces.

In this paper, we study quasinormal and hyponormal composition operators \W with linear fractional compositional symbol $\ph$ on the Hardy and weighted Bergman spaces. We characterize the quasinormal composition operators induced on $H^{2}$ and $A_{\alpha}^{2}$ by these maps and many such weighted composition operators, showing that they are necessarily normal in all known cases. We eliminate several possibilities for hyponormal weighted composition operators but also give new examples of hyponormal weighted composition operators on $H^2$ which are not quasinormal.

In this paper, we estimate essential norms of weighted composition operators and differences of two composition operators on the weighted Bergman spaces in the unit ball.

Let \(\varphi \) be an arbitrary linear-fractional self-map of the unit disk \({\mathbb {D}}\) and consider the composition operator \(C_{-1, \varphi }\) and the Toeplitz operator \(T_{-1,z}\) on the Hardy space \(H^2\) and the corresponding operators \(C_{\alpha , \varphi }\) and \(T_{\alpha , z}\) on the weighted Bergman spaces \(A^2_{\alpha }\) for \(\alpha >-1\). We prove that the unital C\(^*\)-algebra \(C^*(T_{\alpha , z}, C_{\alpha , \varphi })\) generated by \(T_{\alpha , z}\) and \(C_{\alpha , \varphi }\) is unitarily equivalent to \(C^*(T_{-1, z}, C_{-1, \varphi }),\) which extends a known result for automorphism-induced composition operators. For maps \(\varphi \) that are not automorphisms of \({\mathbb {D}}\), we show that \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })\) is unitarily equivalent to \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})\), where \({\mathcal {K}}_{\alpha }\) and \({\mathcal {K}}_{-1}\) denote the ideals of compact operators on \(A^2_{\alpha }\) and \(H^2\), respectively, and apply existing structure theorems for \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})/{\mathcal {K}}_{-1}\) to describe the structure of \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })/\mathcal {K_{\alpha }}\), up to isomorphism. We also establish a unitary equivalence between related weighted composition operators induced by maps \(\varphi \) that fix a point on the unit circle.

If <TEX>${\psi}$</TEX> is analytic on the open unit disk <TEX>$\mathbb{D}$</TEX> and <TEX>${\varphi}$</TEX> is an analytic self-map of <TEX>$\mathbb{D}$</TEX>, the weighted composition operator <TEX>$C_{{\psi},{\varphi}}$</TEX> is defined by <TEX>$C_{{\psi},{\varphi}}f(z)={\psi}(z)f({\varphi}(z))$</TEX>, when f is analytic on <TEX>$\mathbb{D}$</TEX>. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces <TEX>$H^2({\beta})$</TEX>, we prove that if <TEX>$C_{{\psi},{\varphi}}$</TEX> is cohyponormal on <TEX>$H^2({\beta})$</TEX>, then <TEX>${\psi}$</TEX> never vanishes on <TEX>$\mathbb{D}$</TEX> and <TEX>${\varphi}$</TEX> is univalent, when <TEX>${\psi}{\not\equiv}0$</TEX> and <TEX>${\varphi}$</TEX> is not a constant function. Moreover, for <TEX>${\psi}=K_a$</TEX>, where |a| < 1, we investigate normal, cohyponormal and hyponormal weighted composition operators <TEX>$C_{{\psi},{\varphi}}$</TEX>. After that, for <TEX>${\varphi}$</TEX> which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators <TEX>$C_{{\psi},{\varphi}}$</TEX>, when <TEX>${\psi}{\not\equiv}0$</TEX> and <TEX>${\psi}$</TEX> is analytic on <TEX>$\bar{\mathbb{D}}$</TEX>. Finally, we find all normal weighted composition operators which are bounded below.

In this paper, we investigate the spectra of invertible weighted composition operators with automorphism symbols, on Hardy space $H^2(\mathbb{B}_N)$ and weighted Bergman spaces $A_\alpha^2(\mathbb{B}_N)$, where $\mathbb{B}_N$ is the unit ball of the $N$-dimensional complex space. By taking $N=1$, $\mathbb{B}_N=\mathbb{D}$ the unit disc, we also complete the discussion about the spectrum of a weighted composition operator when it is invertible on $H^2(\mathbb{D})$ or $A_\alpha^2(\mathbb{D})$.