proceedings of the american mathematical society Volume 121, Number 4, August 1994 MULTIPLE CANONICAL DECOMPOSITIONS OF FAMILIES OF OPERATORS AND A MODEL OF QUASINORMAL FAMILIES XIMENACATEPILLÁN, MAREKPTAK, AND WACLAW SZYMAÑSKI (Communicated by Palle E. T. Jorgensen) Dedicted to the memory of our friend Domingo Herrero Abstract. A general method of canonical decompositions of several operator- valued functions (operators) is presented. A model of a family of doubly com- muting quasinormal operators is constructed. 1. Introduction The idea of decomposing an operator (a family of operators, an operator- valued function) into parts, which are easier to investigate than the original operator, is fundamental to the theory of operators. The so-called canonical decomposition is one of many known kinds of de- compositions. Wold [11], studying stationary stochastic processes, discovered the decomposition of an isometry into the unitary and the completely nonuni- tary parts, which has since been referred to as the Wold decomposition of an isometry and is the first example of a canonical decomposition. A general theory of canonical decompositions of a single operator (more generally, an operator- valued function) with respect to a single property has been developed in [7]. Following Wold's idea, Slocinski [6] studied canonical decompositions of two commuting isometries. Using in an essential manner the isometric-unitary prop- erties, he established a criterion for "the best" analogue of the Wold decompo- sition to exist and found an ingenious example of two commuting isometries without such decomposition [6, Example 1]. In §2 we present a general method of constructing canonical decompositions of several operator-valued functions (operators, in particular) with respect to several properties, which, in order to make such decompositions possible, not only should behave well themselves (as in [7]) but also should behave well with respect to each other (Theorem 2.2). This multiple canonical decomposition method is based on results of [7]. Canonical decompositions are often the first step in constructing models of operators. Of particular interest here is the model of a single quasinormal Received by the editors November 9, 1992; part of this paper was presented to the 99th Annual Meeting of the AMS, San Antonio, Texas, 1993. 1991 Mathematics Subject Classification. Primary 47A65, 47B20. The first author was partially supported by a grant from Millersville University. ©1994 American Mathematical Society 0002-9939/94 $1.00+ $.25 per page 1165 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use