Extensions of spaces of analytic functions via pointwise limits of bounded sequences and two integral operators on generalized Bloch spaces

Abstract

In [2], operators

$$P_\mu f(z):=-\frac{1}{(1-z)^{\mu+1}} \int \limits_1^z f(\zeta)(1-\zeta)^{\mu} \,d\zeta$$

and

$$Q_\mu f(z):=(1-z)^{\mu-1} \int\limits_0^z f(\zeta)(1-\zeta)^{-\mu} \,d \zeta\quad (z \in \mathbb{D})$$

were investigated in the setting of the analytic Besov spaces B _p , 1 ≤ p ≤ ∞, and the little Bloch space B _∞,0. In particular, for X = B _p , 1 ≤ p < ∞, or X = B _∞,0, the spectra, essential spectra of P _μ , and Q _μ in ${\mathcal {L}(X),}$ together with one sided analytic resolvents in the Fredholm regions of P _μ , and Q _μ were obtained along with an explicit strongly decomposable operator extending Q _μ and simultaneously lifting P _μ . In the current paper, we extend the spectral analysis to generalized Bloch spaces using a modification of a construction due to Aleman and Persson, [3].