Unitary Equivalence of the Unilateral Shift

Abstract

Shifts (including the simple unilateral and bilateral ones) are a basic tool in operator theory. The unilateral shift in particular, has many curious properties, both algebraic and analytic. The techniques for discovering and proving these properties are frequently valuable even when the properties themselves have no visible immediate application. One of the main reasons for the success of the Beurling treatment of the unilateral shift U is that every non-trivial part of U is unitarily equivalent to U. In proving this result Beurling makes use of the Hardy space H2 In this paper we show that every non-trivial part of U is unitarily equivalent to U by using e l(Z) space in place of the Hardy space H' and the connection between wandering subspaces and invariant subspaces. Some isometries are unitary, and some are not; an example of the latter kind is the unilateral shift. More precisely, the direct sum of a unitary operator and a number of copies (finite or infinite) of the unilateral shift is an isometry. It follows that the unilateral shift is more than just an example of an isometry, with interesting and peculiar properties: it is in fact one of the fundamental building blocks out of which all isometries are constructed. Wandering subspaces are important because they are connected with invariant subspaces, in this sense: if U is an isometry, then there is a natural one to one correspondence between all wandering subspaces N and some invariant subspaces M.