Equivalence of Absolute Continuity and Apostol's Condition

Absolute continuity of polynomially bounded $n$-tuples of commuting contractions is studied. A necessary and sufficient condition is found in Constantin Apostol’s “weakened $C_{0,\cdot }$ assumption”, asserting the convergence to 0 of the powers of each operator in a specific topology. Kosiek with Octavio considered tuples of Hilbert space contractions satisfying the von-Neumann Inequality. We extend their results to a wider class of tuples, where there may be no unitary dilation and the bounding constant may be greater than 1. This proof also applies to Banach space contractions and uses decompositions of A-measures with respect to bands of measures related to Gleason parts of the polydisc algebra.