Numerical invariants for commuting isometric pairs

He, Wenping; Qin, Yueshi; Yang, Rongwei

doi:10.1512/iumj.2015.64.5409

Cited by 13 publications

(11 citation statements)

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“…Also note that, in the above theorem, one can choose the coefficient Hilbert space D as ran(I − T T * ) (see [13]). In contrast with the von-Neumann and Wold decomposition theorem for isometries, the structure of commuting n-tuples of isometries, n ≥ 2, is much more complicated and very little, in general, is known (see [3,4,5,6,7,11,12,17,18,15]). However, for pure pairs of commuting isometries, the problem is more tractable.…”

Section: Introductionmentioning

confidence: 99%

Factorizations of contractions

Das

Sarkar

2017

Advances in Mathematics

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The celebrated Sz.-Nagy and Foias theorem asserts that every pure contraction is unitarily equivalent to an operator of the form, for some Hilbert space D. On the other hand, the celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry V on a Hilbert space H is a product of two commuting isometries V 1 and V 2 in B(H) if and only if there exist a Hilbert space E, a unitary U in B(E) and an orthogonal projectionIn this context, it is natural to ask whether similar factorization results hold true for pure contractions. The purpose of this paper is to answer this question. More particularly, let T be a pure contraction on a Hilbert space H and let P Q M z | Q be the Sz.-Nagy and Foias representation of T for some canonical Q ⊆ H 2 D (D). Then T = T 1 T 2 , for some commuting contractions T 1 and T 2 on H, if and only if there exist B(D)-valued polynomials ϕ and ψ of degree ≤ 1 such that Q is a joint (M * ϕ , M * ψ )-invariant subspace,Moreover, there exist a Hilbert space E and an isometry V ∈ B(D; E) such thatwhere the pair (Φ, Ψ), as defined above, is the Berger, Coburn and Lebow representation of a pure pair of commuting isometries on H 2 E (D). As an application, we obtain a sharper von Neumann inequality for commuting pairs of contractions.2010 Mathematics Subject Classification. 47A13, 47A20, 47A56, 47A68, 47B38, 46E20, 30H10.

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Section: Introductionmentioning

confidence: 99%

Factorizations of contractions

Das

Sarkar

2017

Advances in Mathematics

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“…Proof. The equivalence of (a), (b), (c), (d) and (e) follows easily from (1.2) and (1.3), see also [10].…”

Section: Introductionmentioning

confidence: 77%

“…The general case of the positive defect operator. We start with a characterization available in [10] and [12]. The only non-trivial part is the equivalence of (a) and (e).…”

Section: 2mentioning

confidence: 99%

“…We shall first need a result relating positivity of the defect operator C(V 1 , V 2 ) with double commutativity of the pair (V 1 , V 2 ). See also [10] and [12]. Lemma 2.2.…”

Section: Introductionmentioning

confidence: 97%

“…A chief ingredient in our analysis of commuting isometries is the concept of the defect operator C(V 1 , V 2 ) introduced in [9] and [10].…”

Section: Introductionmentioning

confidence: 99%

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

Bhattacharyya¹,

Rastogi²,

U³

2021

Preprint

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The study of a pair (V 1 , V 2 ) of commuting isometries is a classical theme. We shine new light on it by using the defect operator. In the cases when the defect operator is zero or positive or negative, or the difference of two mutually orthogonal projections adding up to ker(V 1 V 2 ) * , we write down structure theorems forThe structure theorems allow us to compute the joint spectra in each of the cases above. Moreover, in each case, we also point out at which stage of the Koszul complex the exactness breaks. With (V 1 , V 2 ), a pair of operator valued functions φ 1 , φ 2 can be canonically associated. In certain cases, ∪ z∈D σ(φ 1 (z), φ 2 (z)) is the same as σ(V 1 , V 2 ) and in certain cases, it is smaller. All such cases are described.A major contribution of this note is to figure out the fundamental pair of commuting isometries with negative defect. We construct this pair of commuting isometries on the Hardy space of the bidisc (It has been known that the fundamental pair of commuting isometries with positive defect is the pair of multiplication operators by the coordinate functions on the Hardy space of the bidisc).

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