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 $H^p(\bb D^n) \ (1\le p\le \infty)$ that remain invariant under the action of coordinate wise multiplication by an $n$-tuple $(T_{B_1},\dots, T_{B_n})$ of operators where each $B_i, \ 1\le i\le n,$ is a finite Blaschke factor on the open unit disc. The critical point to be noted is that these $T_{B_i}$ are assumed to be weaker than isometries as operators. Thus our main theorem extends the principal result of \cite{LS} 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 $B_i, \ 1\le i\le n;$ $(iii)$ from vector subspaces of $H^2(\bb D)$ to the case of vector subspaces of $H^p(\bb D^n), \ 1\le p\le \infty.$ 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.