Invariant subspaces associated with near isometries and unicellular operators

Abstract

The first part of this presentation is devoted to generalizing the famous Beurling s Invariant subspace Theorem for the shift operator to the case of the tuple of operators, where the operators assumed are weaker than isometries, we will be referring to this weaker condition of operators as near-isometries. To begin with, we first derive a generalization of Slocinski s well-known Wold type decomposition of a pair of doubly commuting isometries to the case of an n-tuple of doubly commuting operators near-isometries. Then, with the help of Wold decomposition for the n-tuple of doubly commuting near-isometries, we will represent in concrete fashion those Hilbert spaces that are vector subspaces of the Hardy spaces Hp(Dn) (1 and#8804; p and#8804; and#8734;) that remain invariant under the action of coordinate wise multiplication by an n-tuple (TB1,...,TBn ) of operators where for each 1 and#8804; i and#8804; n, Bi is a finite Blaschke factor on the open unit disc. The critical point to be noted is that these TBi are assumed to be near-isometries. newline newlineThe second part aims to characterize the invariant subspaces of the operator p(T) for some polynomials p, where T is a unicellular forward weighted shift. Here, we study subspaces that are invariant under squares and cubes (separately as well as jointly) of unicellular backward weighted shift operators on a separable Hilbert space. The finite-dimensional subspaces are characterized for all weights and the infinite-dimensional subspaces are characterized for two classes of weights. We will give a sufficient condition on the analytic function f such that the operator f(T) becomes unicellular whenever T is a quasinilpotent unicellular operator. newline