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

Abstract

This paper deals with representing in concrete fashion those Hilbert spaces that are vector subspaces of the Hardy spaces Hp(Dn)(1≤p≤∞) that remain invariant under the action of coordinate wise multiplication by an n-tuple (TB1,…,TBn) of operators where each Bi,1≤i≤n, is a finite Blaschke factor on the open unit disc. The critical point to be noted is that these TBi are assumed to be weaker than isometries 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 Bi,1≤i≤n; (iii) from vector subspaces of H2(D) to the case of vector subspaces of Hp(Dn),1≤p≤∞. We further derive a generalization of Slocinski's well known Wold type decomposition of a pair of doubly commuting isometries to the case of n-tuple of doubly commuting operators whose actions are weaker than isometries.