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

Abstract

We investigate spectral properties of operators of the form $$\begin{aligned} P_\mu f(z):=-\frac{1}{(1-z)^{\mu +1}}\int _1^z f(\zeta )(1-\zeta )^{\mu }\,d\zeta \end{aligned}$$ and $$\begin{aligned} Q_\mu f(z):=(1-z)^{\mu -1}\int _0^z f(\zeta )(1-\zeta )^{-\mu }\,d\zeta \quad (z\in \mathbb{D }) \end{aligned}$$ acting on the analytic Besov spaces \(B_p\) and the little Bloch space \(\mathcal B _0\). For \(X=B_p\), \(1\le p\le \infty \), or \(X=\mathcal B _0\), we identify the spectra of \(P_\mu \) and \(Q_\mu \) in \(\mathcal{L }(X)\), as well as, in the case \(X\ne B_\infty \), the essential spectrum and index together with one sided analytic resolvents in the Fredholm regions of \(P_\mu \) and \(Q_\mu \).