Structure theorems for star-commuting power partial isometries

Abstract

Abstract. We give a new formulation and proof of a theorem of Halmos and Wallen on the structure of power partial isometries on Hilbert space. We then use this theorem to give a structure theorem for a finite set of partial isometries which star-commute: each operator commutes with the others and with their adjoints.

The structure of doubly non-commuting isometries

“…Since we have already observed in Corollary 2.10 that V | H uni is unitary, (1) implies (3). Lemma 2.3 shows that (3) implies (1). The equivalence of (3) and ( 2) is obvious from Proposition 2.1.…”

“…, V n ) of doubly non-commuting isometries are a direct sum of 2 n so-called standard n-tuples. This Wold decomposition for all operators in the n-tuple simultaneously is the statement of Theorem 4.6; if all z ij are equal to 1, this is a particular case of [1,Theorem 2.25]. The 2 n standard n-tuples correspond to the 2 n components in the decomposition of the space from Section 3 as mentioned above.…”

“…In the second part, which is more involved, each of the summands from the first step is written as a Hilbert direct sum of copies of a wandering subspace; see Theorem 3.6. The method to obtain this is not an inductive procedure as in [1,6,8,10], but consists of multiplying n decompositions of the identity operator and interpreting the result.…”

“…In [6] and [8], concerned with general n, no particular terminology is employed. In the case where z ij = 1 for all i and j, [10] speaks of doubly commuting isometries, and [1] of star-commuting (power partial) isometries. Since the noncommuting relation…”

“…We would like to mention explicitly that the method of taking a product of various decompositions of the identity operator differs from inductive approaches as in [1,6,8,10]. Employing such a product may be a more transparent way of working, although this remains a matter of taste.…”