Division by inner functions in a class of composition operators on Lipschitz spaces

In this work we study the essential spectra of composition operators on weighted Bergman spaces of analytic functions which might be termed as "quasi-parabolic." This is the class of composition operators on $A_{\alpha}^{2}$ with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form $\varphi(z)=$ $z+\psi(z)$, where $\psi\in$ $H^{\infty}(\mathbb{H})$ and $\Im(\psi(z)) > \epsilon > 0$. We especially examine the case where $\psi$ is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.

<p>In this article, we considered a class of composition operators on Lebesgue spaces with variable exponents over metric measure spaces. Taking advantage of the compatibility between the metric-measurable structure and the regularity properties of the variable exponent, we provided necessary and sufficient conditions for this class of operators to be bounded and compact, respectively. In addition, we showed the usefulness of the variable change to study weak compactness properties in the framework of non-standard spaces.</p>

Aluthge transforms of weighted shifts on directed trees

The single equality [formula omitted] does not imply the quasinormality of weighted shifts on rootless directed trees

In this paper, we study weighted composition operators between Banach spaces of scalar-valued Lipschitz functions on pointed metric spaces, not necessarily compact. We give necessary and sufficient conditions for the injectivity and the surjectivity of these operators. We also obtain sufficient and necessary conditions for a weighted composition operator between these spaces to be compact.

In this paper, we study quasicompact and Riesz composition operators on Banach spaces of Lipschitz–Holder functions on pointed metric spaces. For a composition operator T on these spaces, we give an upper bound for $$r_e(T)$$ , the essential spectral radius of T, and establish a formula for $$r_e(T)$$ whenever metric spaces are compact. We also give some necessary and some sufficient conditions that a composition operator T on these spaces to be quasicompact or Riesz. Finally, we get a relation for the set of eigenvalues and the spectrum of a quasicompact and Riesz composition operator on these spaces.

We present a non-weak supercyclicity criterion for vectors in infinite dimensional Banach spaces. Also, we give sufficient conditions under which a class of weighted composition operators on a Banach space of analytic functions is not weakly supercyclic. In particular, we show that the semigroup of linear isometries on the spaces $S^p$ ($p>1$), is not weakly supercyclic. Moreover, we observe that every composition operator on some Banach space of analytic functions such as the disc algebra or the analytic Lipschitz space is not weakly supercyclic.

Compact composition operators on generalized Lipschitz spaces