Explicit and Unique Construction of Tetrablock Unitary Dilation in a Certain Case

Bhattacharyya, Tirthankar; Sau, Haripada

doi:10.1007/s11785-015-0472-9

Cited by 6 publications

(14 citation statements)

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“…Fundamental operators ever since its invention [12] have been proved to be of extreme importance in multi-variable dilation theory, see [12,13,14]. The following lemma that follows from Theorem 24 and the remark following it, shows that the fundamental operators are present in both the above constructions of Andô dilation also.…”

Section: Future Researchmentioning

confidence: 92%

“…Then the pairs of operators (F 1 , F 2 ) on D T and (G 1 , G 2 ) on D T * defined by (F 1 , F 2 ) := (Λ * P ⊥ UΛ, Λ * U * P Λ) and (G 1 , G 2 ) := (Γ * P ′⊥ U ′ Γ, Γ * U ′ * P ′ Γ), (6.2) respectively, are the fundamental operators of the tetrablock contractions (T 1 , T 2 , T ) and (T * 1 , T * 2 , T * ), respectively. A normal boundary dilation for the tetrablock was found in [11,14] under the conditions that the fundamental operators…”

Section: Future Researchmentioning

confidence: 99%

“…(III) The idea of the construction of a tetrablock isometric dilation in [11] is invoked in our first construction of Andô dilation. Later in [14] a tetrablock unitary dilation was constructed. On the other hand, if (U 1 , U 2 ) is an Andô unitary dilation of a pair (T 1 , T 2 ) of commuting contractions, then one easily sees that (U 1 , U 2 , U 1 U 2 ) is a tetrablock unitary dilation of (T 1 , T 2 , T 1 T 2 ).…”

Section: Future Researchmentioning

confidence: 99%

“…On the other hand, if (U 1 , U 2 ) is an Andô unitary dilation of a pair (T 1 , T 2 ) of commuting contractions, then one easily sees that (U 1 , U 2 , U 1 U 2 ) is a tetrablock unitary dilation of (T 1 , T 2 , T 1 T 2 ). But one can check that, unfortunately, a similar use of the ideas invoked in [14] does not work for a Schäffer-type construction of Andô unitary dilation. So a Schäffertype construction of Andô unitary dilation still remains open.…”

Section: Future Researchmentioning

confidence: 99%

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Andô dilations for a pair of commuting contractions: two explicit constructions and functional models

2017

Preprint

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One of the most important results in operator theory is Andô's [3] generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Schäffer [29] and Douglas [18]) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an Andô dilation for a commuting pair (T 1 , T 2 ) of contractions, except in some special cases [2,16,17]. In this paper, we give two new proofs of Andô's dilation theorem by giving both Schäffer-type and Douglas-type explicit constructions of an Andô dilation with function-theoretic interpretation, for the general case. The results, in particular, give a complete description of all possible factorizations of a given contraction T into the product of two commuting contractions. Unlike the one-variable case, two minimal Andô dilations need not be unitarily equivalent. However, we show that the compressions of the two Andô dilations constructed in this paper to the minimal dilation spaces of the contraction T 1 T 2 , are unitarily equivalent.In the special case when the product T = T 1 T 2 is pure, i.e., if T * n → 0 strongly, an Andô dilation was constructed recently in [17], which, as this paper will show, is a corollary to the Douglas-type construction. We also show that their construction in this special case can be derived from a previous result obtained in [28].We define a notion of characteristic triple for a pair of commuting contractions and a notion of coincidence for such triples. We prove that two pairs of commuting contractions with their products being pure contractions are unitarily equivalent if and only if their characteristic triples coincide. We also characterize triples which qualify as the characteristic triple for some pair (T 1 , T 2 ) of commuting contractions such that T 1 T 2 is a pure contraction.

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Section: Future Researchmentioning

confidence: 92%

Section: Future Researchmentioning

confidence: 99%

Section: Future Researchmentioning

confidence: 99%

Section: Future Researchmentioning

confidence: 99%

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Andô dilations for a pair of commuting contractions: two explicit constructions and functional models

2017

Preprint

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$\Gamma$-Unitaries, Dilation and a Natural Example

Bhattacharyya¹,

Sau²

2017

Publ. Res. Inst. Math. Sci.

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This note constructs an explicit normal boundary dilation for a commuting pair (S, P ) of bounded operators with the symmetrized bidiskas a spectral set. Such explicit dilations had hitherto been constructed only in the unit disk [11], the unit bidisk [3] and in the tetrablock [6]. The dilation is minimal and unique under a suitable condition. This paper also contains a natural example of a Γ-isometry. We compute its associated fundamental operator.Date: April 4, 2018. MSC2010: Primary:47A20, 47A25.

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Functional Models for Commuting Hilbert-Space Contractions

Ball

2020

Operator Theory, Operator Algebras and Their Interactions With Geometry and Topology

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We develop a Sz.-Nagy-Foias-type functional model for a commutative contractive operator tuple T = (T1, . . . , T d ) having T = T1 • • • T d equal to a completely nonunitary contraction. We identify additional invariants G ♯ , W ♯ in addition to the Sz.-Nagy-Foias characteristic function ΘT for the product operator T so that the combined triple (G ♯ , W ♯ , ΘT ) becomes a complete unitary invariant for the original operator tuple T . For the case d ≥ 3 in general there is no commutative isometric lift of T ; however there is a (not necessarily commutative) isometric lift having some additional structure so that, when compressed to the minimal isometric-lift space for the product operator T , generates a special kind of lift of T , herein called a pseudo-commutative contractive lift of T , which in turn leads to the functional model for T . This work has many parallels with recently developed model theories for symmetrized-bidisk contractions (commutative operator pairs (S, P ) having the symmetrized bidisk Γ as a spectral set) and for tetrablock contractions (commutative operator triples (A, B, P ) having the tetrablock domain E as a spectral set).

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Explicit and Unique Construction of Tetrablock Unitary Dilation in a Certain Case

Cited by 6 publications

References 11 publications

Andô dilations for a pair of commuting contractions: two explicit constructions and functional models

Andô dilations for a pair of commuting contractions: two explicit constructions and functional models

$\Gamma$-Unitaries, Dilation and a Natural Example

Functional Models for Commuting Hilbert-Space Contractions

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