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 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>

We prove that the class 𝒞 ( H Λ α , H Λ β p ) = { ϕ : C ϕ ( H Λ α ) ⊂ H Λ β p } , where HΛα and H Λ β p are holomorphic Lipschitz spaces, has the f-property for p⩾1, 0<β<1: If ϕ I ∈ 𝒞 ( H Λ α , H Λ β p ) where I is an inner function, then ϕ ∈ 𝒞 ( H Λ α , H Λ β p ) Also we characterize inner functions that belong to 𝒞 ( H Λ α , H Λ β p ) for p>0, 0<α⩽1 and 0<β⩽1. The results are deduced from a new, derivative-free, characterization of 𝒞 ( H Λ α , H Λ β p ) that extends some results of Dyakonov [Acta Math. 178 (1997) 143–167.] and the author [Acta Math. 183 (1999) 141–143; J. Math. Anal. Appl., 326 (2007) 1–11].