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The failure of rational dilation on a triply connected domain

Authors: Michael A. Dritschel and Scott McCullough
Journal: J. Amer. Math. Soc. 18 (2005), 873-918
MSC (2000): Primary 47A25; Secondary 30C40, 30E05, 30F10, 46E22, 47A20, 47A48
Published electronically: June 2, 2005
MathSciNet review: 2163865
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Abstract: For $R$ a bounded triply connected domain with boundary consisting of disjoint analytic curves there exists an operator $T$ on a complex Hilbert space $\mathcal H$ so that the closure of $R$ is a spectral set for $T$, but $T$ does not dilate to a normal operator with spectrum in $B$, the boundary of $R$. There is considerable overlap with the construction of an example on such a domain recently obtained by Agler, Harland and Raphael using numerical computations and work of Agler and Harland.

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Additional Information

Michael A. Dritschel
Affiliation: School of Mathematics and Statistics, Merz Court, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, United Kingdom

Scott McCullough
Affiliation: Department of Mathematics, University of Florida, Box 118105, Gainesville, Florida 32611-8105

Keywords: Dilations, spectral sets, multiply connected domains, inner functions, Herglotz representations, Fay reproducing kernels, Riemann surfaces, theta functions, transfer functions, Nevanlinna-Pick interpolation
Received by editor(s): April 28, 2004
Published electronically: June 2, 2005
Additional Notes: The first author’s research was supported by the EPSRC
The second author’s research was supported by the NSF
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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