Biisometric operators and biorthogonal sequences

TL;DR

This paper explores biisometric operators and biorthogonal sequences in Hilbert spaces, revealing their properties, relationships with unilateral shifts, and applications to Laguerre operators on L2.

Contribution

It introduces the concept of biisometric pairs of operators and demonstrates their connection to biorthogonal sequences and Laguerre operators.

Findings

Biisometric pairs share properties with unilateral shifts.

Such pairs generate biorthogonal sequences shifted by the operators.

Applications to Laguerre operators on L2 are demonstrated.

Abstract

It is shown that a pair of Hilbert space operators V and W such that V*W=I (called a biisometric pair) shares some common properties with unilateral shifts when orthonormal basis are replaced with biorthogonal sequences, and it is also shown how such a pair of biisometric operators yields a pair of biorthogonal sequences which are shifted by them. These are applied to a class of Laguerre operators on L2.