Multivariable Sub-Hardy Hilbert Spaces Invariant under the action of $n$-tuple of Finite Blaschke factors

TL;DR

This paper characterizes multivariable sub-Hardy Hilbert spaces invariant under an n-tuple of operators weaker than isometries, extending previous results from single-variable to multivariable cases with finite Blaschke factors.

Contribution

It generalizes the invariant subspace characterization from one variable to multiple variables and from isometries to weaker operators, including a multivariable Wold decomposition.

Findings

Extended the main theorem to n variables and finite Blaschke factors.

Provided a multivariable generalization of Slocinski's Wold decomposition.

Characterized invariant subspaces under weaker operator conditions.

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.