On the Model and Invariant Subspaces for Pairs of Commuting Isometries

Burdak, Zbigniew

doi:10.1007/s00020-019-2516-4

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…The unitary operator U is often regarded as the 'unitary part' of the isometry V . Several attempts have been made to obtain a multivariable analogue of Wold decomposition, see [5,18,22,23] and references therein. Perhaps the most elegant among these models is the one obtained by Berger, Coburn and Lebow [3], see Theorem 2.3.…”

Section: Introductionmentioning

confidence: 99%

Toeplitz operators and pseudo-extensions

Bhattacharyya,

Das,

2019

Preprint

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There are three main results in this paper. First, we find an easily computable and simple condition which is necessary and sufficient for a commuting tuple of contractions to possess a non-zero Toeplitz operator. This condition is just that the adjoint of the product of the contractions is not pure. On one hand this brings out the importance of the product of the contractions and on the other hand, the non-pureness turns out to be equivalent to the existence of a pseudo-extension to a tuple of commuting unitaries. The second main result is a commutant pseudo-extension theorem obtained by studying the unique canonical unitary pseudo-extension of a tuple of commuting contractions. The third one is about the C * -algebra generated by the Toeplitz operators determined by a commuting tuple of contractions. With the help of a special completely positive map, a different proof of the existence of the unique canonical unitary pseudo-extension is given.

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Section: Introductionmentioning

confidence: 99%

Toeplitz operators and pseudo-extensions

Bhattacharyya,

Das,

2019

Preprint

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“…Thus, for n ≥ 3 history witnessed only conditional unitary dilations for commuting contractions. Some notable works in this direction are due to Agler [1], Brehmer [17], Curto, Vailecu [20,21], Ball, Li, Timotin, Trent [9], Ball, Trent, Vinnikov [10], Popescu [43,44,45], Eschmeier [26], M üller, Vailecu [34], M üller, Ptak [35], Bhat, Bhattacharyya, Dey [16], Burdak [18], Barik, Das, Sarkar [11] and many others, see the reference list and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Minimal unitary dilations for commuting contractions

Pal¹,

Sahasrabuddhe²

2022

Preprint

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acting on a Hilbert space H with T = ∏ n i=1 T i , we show that (T 1 , . . . , T n ) dilates to commuting isometries (V 1 , . . . ,V n ) on the minimal isometric dilation space of T with V = ∏ n i=1 V i being the minimal isometric dilation of T if and only ifi=1 Y i being the minimal isometric dilation of T * . Then, we prove an analogue of this result for unitary dilations of (T 1 , . . . , T n ) and its adjoint. We find a necessary and sufficient condition such that (T 1 , . . . , T n ) possesses a unitary dilation (W 1 , . . . ,W n ) on the minimal unitary dilation space of T with W = ∏ n i=1 W i being the minimal unitary dilation of T . We show an explicit construction of such a unitary dilation on both Sch äffer and Sz. Nagy-Foias minimal unitary dilation spaces of T . Also, we show that a relatively weaker hypothesis is necessary and sufficient for the existence of such a unitary dilation when T is a C. 0 contraction, i.e. when T * n → 0 strongly as n → ∞. We construct a different unitary dilation for (T 1 , . . . , T n ) when T is a C. 0 contraction.

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“…There has been numerous generalizations of this classical decomposition theorem. For example, see [2,22] for development in the commutative setting and [19,23] for doubly commutative setting; also see [4,5,8,14,15,16,17,24] and references therein for results in this direction.…”

Section: Introductionmentioning

confidence: 99%

Models for q-commutative tuples of isometries

Ball¹,

Sau²

2022

Preprint

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A pair of Hilbert space linear operators (V 1 , V 2 ) is said to be q-commutative, for a unimodular complex number qA concrete functional model for q-commutative pairs of isometries is obtained. The functional model is parametrized by a collection of Hilbert spaces and operators acting on them. As a consequence, the collection serves as a complete unitary invariance for q-commutative pairs of isometries. A q-commutative operator pair (V 1 , V 2 ) is said to be doubly q-commutative, if in addition, it satisfies V 2 V * 1 = qV * 1 V 2 . Doubly q-commutative pairs of isometries are also characterized. Special attention is given to doubly q-commutative pairs of shift operators. The notion of q-commutativity is then naturally extended to the case of general tuples of operators to obtain a similar model for tuples of q-commutative isometries.

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On the Model and Invariant Subspaces for Pairs of Commuting Isometries

Cited by 5 publications

References 15 publications

Toeplitz operators and pseudo-extensions

Toeplitz operators and pseudo-extensions

Minimal unitary dilations for commuting contractions

Models for q-commutative tuples of isometries

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