The failure of rational dilation on a triply connected domain

Dritschel, Michael A.; McCullough, Scott

doi:10.1090/s0894-0347-05-00491-1

Cited by 63 publications

(35 citation statements)

References 27 publications

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“…For T to have a normal ∂X-dilation it is necessary that X be a spectral set for T . Sufficiency has been investigated for many domains in C and several interesting results are known including success of such a dilation on an annulus( [1]) and its failure in triply connected domains( [2], [18]). When (T 1 , T 2 ) is a commuting pair of operators for which D 2 is a spectral set, Ando's theorem provides a simultaneous commuting unitary dilation of (T 1 , T 2 ).…”

Section: Introductionmentioning

confidence: 99%

A functional model for pure $\Gamma$-contractions

Bhattacharyya¹,

Pal²

2014

J. Operator Theory

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A pair of commuting operators (S, P ) defined on a Hilbert space H for which the closed symmetrized bidiscis a spectral set is called a Γ-contraction in the literature. A Γ-contraction (S, P ) is said to be pure if P is a pure contraction, i.e, P * n → 0 strongly as n → ∞. Here we construct a functional model and produce a set of unitary invariants for a pure Γ-contraction. The key ingredient in these constructions is an operator, which is the unique solution of the operator equationand is called the fundamental operator of the Γ-contraction (S, P ). We also discuss some important properties of the fundamental operator.

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Section: Introductionmentioning

confidence: 99%

A functional model for pure $\Gamma$-contractions

Bhattacharyya¹,

Pal²

2014

J. Operator Theory

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mentioning

confidence: 99%

Dilations of Γ-contractions by solving operator equations

Bhattacharyya

Pal

Roy

2012

Advances in Mathematics

113

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“…We would also like to mention the following deep result due to Agler [1]: if an annulus X is a spectral set for A, then it is a complete spectral set for A, that is, A has a normal ∂X-dilation. However, the analogue of Agler's result is not true for triply connected domains (see [2,10,18]).…”

Section: Shields' Question For the Annulus Let R > 1 Given An Inverti...mentioning

confidence: 96%

Intersections of several disks of the Riemann sphere as $K$-spectral sets

Badea

Beckermann

Crouzeix

2009

Communications on Pure &Amp; Applied Analysis

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We prove that if n closed disks D 1 , D 2 ,. . . , Dn, of the Riemann sphere are spectral sets for a bounded linear operator A on a Hilbert space, then their intersection D 1 ∩ D 2 · · · ∩ Dn is a complete K-spectral set for A, with K ≤ n + n(n − 1)/ √ 3. When n = 2 and the intersection X 1 ∩ X 2 is an annulus, this result gives a positive answer to a question of A.L. Shields (1974). 1 2 CATALIN BADEA, BERNHARD BECKERMANN, AND MICHEL CROUZEIX means that ρ A cb ≤ K. A complete spectral set is a complete K-spectral set with K = 1. Complete K-spectral sets are important in several problems of Operator Theory (see [16]).Spectral sets were introduced and studied by J. von Neumann [14] in 1951. In the same paper von Neumann proved that a closed disk {z ∈ C : |z − α| ≤ r} is a spectral set for A if and only if A − αI ≤ r. Also, the closed set {z ∈ C : |z − α| ≥ r} is spectral for A ∈ L(H) if and only if (A − αI) −1 ≤ r −1 , and the half-plane {Re(z) ≥ 0} is a spectral set if and only if Re Av, v ≥ 0 for all v ∈ H. Therefore for any closed disk D of the Riemann sphere (interior/exterior of a disk or a half-plane) it is easy to check whether D is a spectral set.We refer to [19] for a treatment of the von Neumann's theory of spectral sets and to [15] and the book [16] for modern surveys of known properties of K-spectral and complete K-spectral sets.The problem. Let X be the intersection of n disks of the Riemann sphere, each of them being a spectral set for a given operator A ∈ L(H). In the present paper we will be concerned with the question whether X itself is a (complete) K-spectral set for A.The intersection of two spectral sets is not necessarily a spectral set; counterexamples for the annulus are presented in [24,13,15]. However, the same question for K-spectral sets remains open. The counterexamples for spectral sets show that the same constant cannot be used for the intersection.Some cases of the K-spectral set problem have been solved. If two K-spectral sets have disjoint boundaries, then by a result of Douglas and Paulsen [9] the intersection is a K ′ -spectral set for some K ′ . The case when the boundaries meet was considered by Stampfli [21,22] and Lewis [12]. In particular, it is proved in [12] that the intersection of a (complete) K-spectral set for the bounded linear operator A with the closure of any open set containing the spectrum of A is a (complete) K ′ -spectral set for A.The main result. In this paper we will show the following result, which gives a positive answer to a question raised by Michael A. Dritschel (personal communication). Theorem 1.1. Let A ∈ L(H), and consider the intersection X = D 1 ∩ D 2 ∩ · · · ∩ D n of n disks of the Riemann sphere C, each of them being spectral for A. Then X is a complete K-spectral set for A, with a constant K ≤ n + n(n−1)/ √ 3.

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The failure of rational dilation on a triply connected domain

Cited by 63 publications

References 27 publications

A functional model for pure $\Gamma$-contractions

A functional model for pure $\Gamma$-contractions

Dilations of Γ-contractions by solving operator equations

Intersections of several disks of the Riemann sphere as $K$-spectral sets

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