On a decomposition for pairs of commuting contractions

Abstract

Abstract.A new decomposition of a pair of commuting, but not necessarily doubly commuting contractions is proposed. In the case of power partial isometries a more detailed decomposition is given.

Structure theorems for star-commuting power partial isometries

“…This last result seems to be new, perhaps even for operators on a finite-dimensional space, and for isometries it looks quite different from the existing versions. For pairs of power partial isometries, it also looks quite different from the decomposition in [2], and the tensor-product decompositions obtained in [4, §3], which are for special cases where the individual Halmos-Wallen decompositions have a single summand, follow from our result.…”

“…It has been known for many years that the most satisfactory results are those for families which star-commute, in the sense that each isometry commutes with the other isometries and with their adjoints (see [13,3,12], and the extensive references in [12]). There have been similar results for pairs of starcommuting power partial isometries based on the Halmos-Wallen theorem [4,2].…”

On the unitary part of isometries in commuting, completely non doubly commuting pairs

“…We solve the problem by other way. Recall form [6], that a pair of commuting contractions V 1 , V 2 is called strongly completely non unitary if there is no proper subspace reducing V 1 , V 2 and at least one of them to a unitary operator. Moreover, there is a decomposition theorem ( [6] The theorem in case of a pair of commuting isometries leads us to the following decomposition.…”

On the commuting isometries

“…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.…”