The canonical Wold decomposition of commuting isometries with finite dimensional wandering spaces

On the Wold-type decompositions for n-tuples of commuting isometric semigroups

“…Since the sum of these projections is the identity operator on H, the conclusion follows. Now, it is obvious that the following result holds: The existence of a multiple decomposition for an n-tuple of commuting isometric semigroups {V i (σ i )} σ i ∈S i , i ∈ I n , may be concluded by various properties like: doubly commutativity, hyperreducivity of the Lebesgue decomposition, finite dimensional wandering spaces ( [2], [3], [20], [25]). Let k ∈ I n , 2 ≤ k < n. We denote In the sequel, two results about a pair of doubly commuting isometric semigroups are presented.…”

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

“…Let us recall the notion of multiple canonical von Neumann-Wold decompositions introduced in [7] in the general case, here taking a simplified form in the case of a pair of commuting isometries: By [4,16] there are multiple canonical von Neumann-Wold decompositions in either of the two cases: for doubly commuting pairs of isometries and for pairs satisfying the conditions dim(ker V * 1 ) < ∞ and dim(ker V * 2 ) < ∞. Moreover, in the case of doubly commuting isometries, the subspace ker V * 1 ∩ ker V * 2 is wandering for the semigroup generated by V 1 , V 2 .…”

“…A natural question arises about generalizations for pairs or families of operators. 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]).…”

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.…”