On decomposition of pairs of commuting isometries

Abstract

Abstract.A review of known decompositions of pairs of isometries is given. A new, finer decomposition and its properties are presented. Introduction. Let

On a decomposition for pairs of commuting contractions

“…A natural example of a commuting, but not doubly commuting pair of isometries is T, T 2 , where T ∈ L(H) is a unilateral shift. For more examples see [1].…”

Shift-Type Properties of Commuting, Completely Non Doubly Commuting Pairs of Isometries

“…The construction of any maximal bilateral shift subspace can be done by considering a maximal wandering subspace. We follow the idea of wandering vectors from [5]. Let G be a semigroup and {T g } g∈G be a semigroup of isometries on H. The vector x ∈ H is called a wandering vector (for a given semigroup of isometries) if for any g 1 = g 2 we have T g1 x, T g2 x = 0.…”

“…Therefore we reduce our attention to completely non doubly commuting pairs of isometries. Such pairs are a special case of a weak bi-shift class whose finer, but not fully satisfying decomposition has been described in [5].…”

“…The most natural generalization, which following [7] is proposed to be called a multiple canonical von Neumann-Wold decomposition, has been achieved only in some special cases ( [4,16]). In the general case various von NeumannWold type decompositions or models were established ( [1][2][3]5,9,10,14,17]). Recall that operators T 1 , T 2 ∈ L(H) doubly commute if they commute and T * 1 T 2 = T 2 T * 1 .…”

On the decomposition of families of quasinormal operators

“…Restrictions T 1 | Hni , T 2 | Hni form a completely non isometric pair. The commuting isometries are being studied by many authors (see [1,3,8]). In the following part we consider a pair of commuting quasinormal partial isometries and describe how one of them acts between the kernel and the isometric part of the other one.…”