Taylor Functional Calculus

Müller, Veronika

doi:10.1007/978-3-0348-0667-1_61

Cited by 4 publications

(6 citation statements)

References 20 publications

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“…is not Taylor regular. For an explicit description of Taylor joint spectrum in general setting, i.e, of an n-tuple of commuting bounded operators one can see [32,33,25].…”

Section: Taylor Joint Spectrum and Preliminary Results Aboutmentioning

confidence: 99%

Subvarieties of the tetrablock and von Neumann's inequality

Pal¹

2014

Preprint

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We show an interplay between the complex geometry of the tetrablock E and the commuting triples of operators having E as a spectral set. We prove that E being a 3-dimensional domain does not have any 2-dimensional distinguished variety, every distinguished variety in the tetrablock is one-dimensional and can be represented as] and a norm condition. The converse also holds, i.e, a set of the form (0.1) is always a distinguished variety in E. We show that for a triple of commuting operators Υ = (T1, T2, T3) having E as a spectral set, there is a one-dimensional subvariety ΩΥ of E depending on Υ such that von-Neumann's inequality holds, i.e, f (T1, T2, T3) ≤ supfor any holomorphic polynomial f in three variables, provided that T n 3 → 0 strongly as n → ∞. The variety ΩΥ has been shown to have representation like (0.1), where A1, A2 are the unique solutions of the operator equations. We also show that under certain condition, ΩΥ is a distinguished variety in E. We produce an explicit dilation and a concrete functional model for such a triple (T1, T2, T3) in which the unique operators A1, A2 play the main role. Also, we describe a connection of this theory with the distinguished varieties in the bidisc and in the symmetrized bidisc.

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“…is not Taylor regular. For an explicit description of Taylor joint spectrum in general setting, i.e, of an n-tuple of commuting bounded operators one can see [32,33,25].…”

Section: Taylor Joint Spectrum and Preliminary Results Aboutmentioning

confidence: 99%

Subvarieties of the tetrablock and von Neumann's inequality

Pal¹

2014

Preprint

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“…For application in Section 11, we don't require it to be so. However, that the two calculi coincide follows from results of M. Putinar [16] (or of V. Müller [15]).…”

Section: Taylor's Holomorphic Functional Calculus and Direct Integralsmentioning

confidence: 94%

“…, r(n)}. By repeated application of the formula (15) proved for rectangles, we get k j=1 P (T : R (j) ) = (q1,...,qn)∈Q…”

Section: Meets and Joins Of Haagerup-schultz Projections Of Commuting...mentioning

confidence: 98%

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Simultaneous Upper Triangular Forms for Commuting Operators in a Finite von Neumann Algebra

Charlesworth¹,

Dykema²,

Sukochev³

et al. 2019

Can. J. Math.-J. Can. Math.

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The joint Brown measure and joint Haagerup-Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved and the joint Brown measure and joint Haagerup-Schultz projections are shown to behave well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.

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“…In fact it is sufficient for the functions f and g to be defined and holomorphic in a neighbourhood of X = Z(h) ∩ Sp(A). Using the methods of [Mül02], Taylor holomorphic calculus can be extended to define operators f (A) and g(A) acting on the Koszul complex K(h, H ) and the conclusions of the theorem will still hold.…”

Section: Introductionmentioning

confidence: 99%

Tate tame symbol and the joint torsion of commuting operators

Kaad,

Nest

2014

Preprint

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We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can be computed by a local formula which involves a tame symbol of the involved holomorphic functions. As an application we are able to extend the classical tame symbol of meromorphic functions on a Riemann surface to the more involved setting of transversal functions on a complex analytic curve. This follows by spelling out our main result in the case of Toeplitz operators acting on the Hardy space over the polydisc. Contents1. Introduction 1 2. Determinants, torsion and joint torsion 6 2.1. Determinants of vector spaces 6 2.2. Fredholm complexes 9 2.3. Joint torsion 10 3. Joint torsion of commuting operators 11 3.1. The Koszul complex 11 3.2. Joint torsion transition numbers 13 3.3. Local indices 14 4. Multiplicative Lefschetz numbers 15 4.1. Lefschetz numbers of Koszul complexes 15 5. Localization of the joint torsion 17 6. Application: Tame symbols of complex analytic curves 21 6.1. Preliminaries on complex analytic spaces 21 6.2. The Tate tame symbol 22 References 25

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Taylor Functional Calculus

Abstract: Abstract. We give a Martinelli-Vasilescu type formula for the Taylor functional calculus and a simple proof of its basic properties.

Cited by 4 publications

References 20 publications

Subvarieties of the tetrablock and von Neumann's inequality

Subvarieties of the tetrablock and von Neumann's inequality

Simultaneous Upper Triangular Forms for Commuting Operators in a Finite von Neumann Algebra

Tate tame symbol and the joint torsion of commuting operators

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