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Unitary Invariants in Multivariable Operator Theory
Gelu Popescu, University of Texas at San Antonio, TX

Memoirs of the American Mathematical Society
2009; 91 pp; softcover
Volume: 200
ISBN-10: 0-8218-4396-6
ISBN-13: 978-0-8218-4396-3
List Price: US$69
Individual Members: US$41.40
Institutional Members: US$55.20
Order Code: MEMO/200/941
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This paper concerns unitary invariants for \(n\)-tuples \(T:=(T_1,\ldots, T_n)\) of (not necessarily commuting) bounded linear operators on Hilbert spaces. The author introduces a notion of joint numerical radius and works out its basic properties. Multivariable versions of Berger's dilation theorem, Berger-Kato-Stampfli mapping theorem, and Schwarz's lemma from complex analysis are obtained. The author studies the joint (spatial) numerical range of \(T\) in connection with several unitary invariants for \(n\)-tuples of operators such as: right joint spectrum, joint numerical radius, euclidean operator radius, and joint spectral radius. He also proves an analogue of Toeplitz-Hausdorff theorem on the convexity of the spatial numerical range of an operator on a Hilbert space, for the joint numerical range of operators in the noncommutative analytic Toeplitz algebra \(F_n^\infty\).

Table of Contents

  • Introduction
  • Unitary invariants for \(n\)-tuples of operators
  • Joint operator radii, inequalities, and applications
  • Bibliography
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