Multivariable sub-Hardy Hilbert spaces invariant under the action of n-tuple of finite Blaschke factors

This paper deals with representing in concrete fashion those Hilbert spaces that are vector subspaces of the Hardy spaces H-p (D-n) (1 <= p <= infinity) that remain invariant under the action of coordinate wise multiplication by an n-tuple (T-B(1), T-B(n)) of operators where each B-i, 1 <= i <= n, is a finite Blaschke factor on the open unit disc. The critical point to be noted is that these T-Bi are assumed to be weaker than isometrics as operators. Thus our main theorems extends the principal result of [10] in the following three directions: (i) from one to several variables; (ii) from multiplication with the coordinate function z to an n-tuple of multiplication by finite Blaschke factors B-i, 1 <= i <= n; (iii) from vector subspaces of H-2 (D) to the case of vector subspaces of H-p(D-n), 1 <= p <= infinity. We further derive a generalization of Slocinski's well known Wold type decomposition of a pair of doubly commuting isometrics to the case of n-tuple of doubly commuting operators whose actions are weaker than isometries. (C) 2022 Published by Elsevier Inc.