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

We investigate spectral properties of operators of the form and 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) acting on the analytic Besov spaces Bp and the little Bloch space B-0. For X = B-p, 1 <= p <= 8, or X = B-0, we identify the spectra of P mu and Q mu in L( X), as well as, in the case X not equal B-infinity, the essential spectrum and index together with one sided analytic resolvents in the Fredholm regions of P-mu and Q(mu).