On the commuting isometries

Abstract

The paper summarizes, clarifies and supplements results on decompositions and a model for pairs of commuting isometries. The geometrical model is given whenever it is possible. The remaining problem of describing the model for completely non-compatible pairs of isometries is reduced to completely non-compatible pairs of unilateral shifts. Moreover, subspaces generating incompatibility are determined.

The decomposition theorems in Baer $*$-rings

“…An example of compatible, completely non doubly commuting pair is (x, x) where x is a non unitary isometry. Another examples can be found in papers on operators on Hilbert spaces [4,5,8]. In particular in [5] there is given a precise model of a commuting, compatible pair.…”

Invariant subspaces of $\mathcal{H}^2(\mathbb{T}^2)$ and $L^2(\mathbb{T}^2)$ preserving compatibility

“…The idea of pairs given by a diagram appeared in [9] (Example 1) while the precise definition and classification of diagrams can be found in [4].…”

“…For purposes of the paper it is convenient to define a pair of isometries given by an arbitrary diagram J by giving the model based on L w , L z . Therefore, instead of Definition 4.3 from [4] it is given another one. First step is to reformulate Definition 4.4 from [4] as follows: Definition 2.2.…”

On the structure and the joint spectrum of a pair of commuting isometries

“…Over several decades Marek S lociński, first by himself and then with his collaborators, has developed a complete structure theorem on commuting pairs of isometries. The most general result is in [5]. One of his early results is the following.…”

“…This note is about the structure (in the style of Wold decomposition) and Taylor joint spectrum of a pair of commuting isometries (V 1 , V 2 ). The topic of commuting isometries has been vigorously pursued in the last two decades, see [1], [3], [4], [5], [11], [12] and [15] for results on structure theory. There has been none on joint spectra.…”