Compatible pairs of commuting isometries

Abstract

The paper is devoted to pairs of commuting isometries. A unique decomposition of such pairs into a compatible and a completely non-compatible part is constructed. The compatible part is fully described.

On the commuting isometries

“…The functional model for compatible pairs of isometries was given in [14], while a more general geometrical one is given in [8]. Let us recall some results from [8].…”

“…Since the model for doubly commuting pairs of isometries is known, it is enough to construct a model for completely non-doubly commuting, compatible pairs. In [8] it is shown that such pairs can be decomposed into two types: pairs defined by a diagram and generalized powers.…”

“…The second type of compatible pairs of isometries is a class of generalized powers defined in [8]. We precede the formal definition by giving some properties and background of the idea of generalized powers.…”

“…For respective wandering vectors such a pair turns out to be a pair of generalized powers but also a pair defined by a diagram. However, such a diagram has to be periodic in the following sense [8]: The idea of a periodic diagram and its period is explained in the following picture:…”

“…Wold's model does not extend to multiple cases easily. Models of systems of isometries were investigated by many mathematicians, like: Charles A. Berger, Lewis A. Coburn, Arnold Lebow [2], Ion Suciu [25], Marek Słociński [23], Karel Horák, Vladimir Müller [13,14], Dimitru Gaşpar, Pǎstorel Gaşpar, Nicolae Suciu [11,12], Ximena Catepillán, Marek Ptak, Wacław Szymański [10] (more general approach), Marek Kosiek and Alfredo Octavio [15], Dan Popovici [19][20][21], Hari Bercovici, Ron Douglas, Ciprian Foiaş [3,4], Jaydeb Sarkar [22] and also the authors of the paper (all or in part) have some contribution [5][6][7][8][9]18]. The above list is by no means exhaustive and can not be treated as a review of the results.…”