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

In [2], operators P(mu)f(z) := -1/(1-z)(mu+1) integral(z)(1) f(zeta)(1-zeta)(mu) d zeta and Q(mu)f(z) := (1-z)(mu-1) integral(z)(0) f(zeta)(1-zeta)(-mu) d zeta (z is an element of D) were investigated in the setting of the analytic Besov spaces B-p, 1 <= p <=infinity, and the little Bloch space B-infinity,B-0. In particular, for X = B-p, 1 <= p<infinity, or X = B-infinity,B-0, the spectra, essential spectra of P-mu, and Q(mu) in L(X), together with one sided analytic resolvents in the Fredholm regions of P-mu , and Q(mu) were obtained along with an explicit strongly decomposable operator extending Q(mu) and simultaneously lifting P-mu . 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].