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

DOI:
https://doi.org/10.1090/S0894-0347-05-00491-1

Published electronically:
June 2, 2005

MathSciNet review:
2163865

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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*A monograph on function theory and Herglotz formulas for multiply connected domains*. AHR Jim Agler, John Harland, and Benjamin Raphael,

*Classical function theory, operator dilation theory, and machine computations on multiply-connected domains*.

*Private communication*.

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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

Email:
m.a.dritschel@ncl.ac.uk

**Scott McCullough**

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

MR Author ID:
220198

Email:
sam@math.ufl.edu

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.