In this paper we estimate the norm and the essential norm of weighted composition operators from a large class of-non-necessarily reflexive-Banach spaces of analytic functions on the open unit disk into weighted type Banach spaces of analytic functions and Bloch type spaces. We also show the equivalence of compactness and weak compactness of weighted composition operators from these weighted type spaces into a class of Banach spaces of analytic functions, that includes a large family of conformally invariant spaces like BMOA and analytic Besov spaces.
We characterize boundedness and compactness of the classical Volterra operator Tg : H ∞ vα → H ∞ induced by a univalent function g for standard weights vα with 0 ≤ α < 1, partly answering an open problem posed by A. Anderson, M. Jovovic and W. Smith. We also study boundedness, compactness and weak compactness of the generalized Volterra operator T ϕ g mapping between Banach spaces of analytic functions on the unit disc satisfying certain general conditions.2010 Mathematics Subject Classification. Primary 47B38, Secondary 46B50.
To Richard Aron on the occasion of his retirement, with very much appreciation
Keywords: Composition operator Königs eigenfunctionWe discuss when the Königs eigenfunction associated with a non-automorphic selfmap of the complex unit disc that fixes the origin belongs to Banach spaces of holomorphic functions of Bloch and H ∞ type. In the latter case, our characterization answers a question of P. Bourdon. Some spectral properties of composition operators on H ∞ for unbounded Königs eigenfunction are obtained.
Abstract. The spectra of invertible weighted composition operators uCϕ on the Bloch and Dirichlet spaces are studied. In the Bloch case we obtain a complete description of the spectrum when ϕ is a parabolic or elliptic automorphism of the unit disc. In the case of a hyperbolic automorphism ϕ, exact expressions for the spectral radii of invertible weighted composition operators acting on the Bloch and Dirichlet spaces are derived.
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