Spectral Properties of Two Classes of Averaging Operators on the Little Bloch Space and the Analytic Besov Spaces

摘要

We investigate spectral properties of operators of the form P_μ f(z) : =- 1/((1-z)~(μ+1))) ∫~z_1 f(ζ)(1-ζ)~μ dζ and Q_μf(z) := (1 - z)~(μ-1) ∫~z_0 f(ζ)(1 - ζ)~μ dζ (z ∈ D) acting on the analytic Besov spaces B_p and the little Bloch space B_0. For X = B_p, 1 ≤ p ≤ ∞, or X = B_0, we identify the spectra of P_μ and Q_μ in ζ(X), as well as, in the case X ≠ B_∞, the essential spectrum and index together with one sided analytic resolvents in the Fredholm regions of P_μ, and Q_μ.