Characterizations of Symmetrized Polydisc

Gorai, Sushil; Sarkar, Jaydeb

doi:10.48550/arxiv.1503.03473

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“…Part (1) ⇔ (4) of Theorem ?? and part (1) ⇔ (7) of Theorem 3.5 were obtained independently by Costara in [22] and later by Gorai and Sarkar, [33]. Also parts of Theorem 5.2, Theorem 5.5 and Theorem 5.3 were proved by Biswas and Shyam Roy [20], in a more general setting.…”

Section: Introductionmentioning

confidence: 86%

“…The symmetrized polydiscs in several dimensions have attracted considerable attention in past two decades because of its rich function theory [3,10,11,23,37,38,40,42,48], complex geometry [9,22,30,33,39,41], associated operator theory [6,8,7,18,19,20,43,45,54] and its connection with the most appealing and difficult problem of µ-synthesis [4,16,61], which arises in the H ∞ approach to the problem of robust control [28,29]. Operator theory on the symmetrized bidisc has numerous applications to its complex geometry and function theory, see classic [5].…”

Section: Introductionmentioning

confidence: 99%

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Rational dilation on the symmetrized tridisc: failure, success and unknown

Pal¹

2016

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The closed symmetrized tridisc Γ3 and its distinguished boundary bΓ3 are the setsA triple of commuting operators (S1, S2, P ) defined on a Hilbert space H for which Γ3 is a spectral set is called a Γ3-contraction. In this article we show by a counter example that there are Γ3-contractions which do not dilate. It is also shown that under certain conditions a Γ3-contraction can have normal bΓ3 dilation. We determine several classes of Γ3-contractions which dilate and show explicit construction of their dilations. A concrete functional model is provided for the Γ3contractions which dilate. Various characterizations for Γ3-unitaries and Γ3-isometries are obtained; the classes of Γ3-unitaries and Γ3-isometries are analogous to the unitaries and isometries in one variable operator theory. Also we find out a model for the class of pure Γ3-isometries. En route we study the geometry of the sets Γ3 and bΓ3 and provide variety of characterizations for them. Contents 1. Introduction 2 1.1. Motivation 2 1.2. Literature and a brief description of the main results 4 2. Spectral set, complete spectral set and rational dilation 8 2.1. The Taylor joint spectrum 8 2.2. Spectral and complete spectral set 9 2.3. The distinguished boundary and rational dilation 9 3. Schur's criterion and geometry of the symmetrized tridisc 10

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Section: Introductionmentioning

confidence: 86%