Journal of the American Mathematical Society

Author(s): Michael A. Dritschel; Scott McCullough
Journal: J. Amer. Math. Soc. 18 (2005), 873-918.
MSC (2000): Primary 47A25; Secondary 30C40, 30E05, 30F10, 46E22, 47A20, 47A48
Posted: June 2, 2005
Retrieve article in: PDF

Abstract: For

a bounded triply connected domain with boundary consisting of disjoint analytic curves there exists an operator

on a complex Hilbert space $\mathcal H$ so that the closure of

is a spectral set for

, but

does not dilate to a normal operator with spectrum in

, the boundary of

. 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.

1.: M. B. Abrahamse and R. G. Douglas, A class of subnormal operators related to multiply-connected domains, Advances in Math. 19 (1976), no. 1, 106-148. MR 0397468 (53:1327)
2.: Jim Agler, Rational dilation on an annulus, Ann. of Math. (2) 121 (1985), no. 3, 537-563. MR 0794373 (87a:47007)
3.: -, On the representation of certain holomorphic functions defined on a polydisc, Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl., vol. 48, Birkhäuser, Basel, 1990, pp. 47-66. MR 1207393 (93m:47013)
4.: Jim Agler and John Harland, A monograph on function theory and Herglotz formulas for multiply connected domains.
5.: Jim Agler, John Harland, and Benjamin Raphael, Classical function theory, operator dilation theory, and machine computations on multiply-connected domains.
6.: Jim Agler and John E. McCarthy, Nevanlinna-Pick interpolation on the bidisk, J. Reine Angew. Math. 506 (1999), 191-204. MR 1665697 (2000a:47034)
7.: -, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259 (2003b:47001)
8.: William Arveson, Subalgebras of $C\sp{\ast}$ -algebras. II, Acta Math. 128 (1972), no. 3-4, 271-308. MR 0394232 (52:15035)
9.: William B. Arveson, Subalgebras of $C\sp{\ast}$ -algebras, Acta Math. 123 (1969), 141-224. MR 0253059 (40:6274)
10.: Joseph A. Ball and Kevin F. Clancey, Reproducing kernels for Hardy spaces on multiply connected domains, Integral Equations Operator Theory 25 (1996), no. 1, 35-57. MR 1386327 (97f:46042)
11.: Joseph A. Ball and Tavan T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal. 157 (1998), no. 1, 1-61. MR 1637941 (2000b:47028)
12.: Joseph A. Ball, Tavan T. Trent, and Victor Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Operator theory and analysis (Amsterdam, 1997), Oper. Theory Adv. Appl., vol. 122, Birkhäuser, Basel, 2001, pp. 89-138. MR 1846055 (2002f:47028)
13.: Joseph A. Ball and Victor Vinnikov, Hardy spaces on a finite bordered Riemann surface, multivariable operator model theory and Fourier analysis along a unimodular curve, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 37-56. MR 1882690 (2003f:47016)
14.: Stefan Bergman and Bruce Chalmers, A procedure for conformal mapping of triply-connected domains, Math. Comp. 21 (1967), 527-542. MR 0228663 (37:4243a)
15.: Kevin F. Clancey, Toeplitz operators on multiply connected domains and theta functions, Contributions to operator theory and its applications (Mesa, AZ, 1987), Oper. Theory Adv. Appl., vol. 35, Birkhäuser, Basel, 1988, pp. 311-355. MR 1017675 (91f:47038)
16.: -, Representing measures on multiply connected planar domains, Illinois J. Math. 35 (1991), no. 2, 286-311. MR 1091446 (92e:46110)
17.: John B. Conway, Functions of one complex variable, second ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York, 1978. MR 0503901 (80c:30003)
18.: -, The theory of subnormal operators, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR 1112128 (92h:47026)
19.: H. M. Farkas and I. Kra, Riemann surfaces, second ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765 (93a:30047)
20.: John D. Fay, Theta functions on Riemann surfaces, Springer-Verlag, Berlin, 1973, Lecture Notes in Mathematics, Vol. 352. MR 0335789 (49:569)
21.: Stephen D. Fisher, Function theory on planar domains, Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1983, A second course in complex analysis, A Wiley-Interscience Publication. MR 0694693 (85d:30001)
22.: Helmut Grunsky, Lectures on theory of functions in multiply connected domains, Vandenhoeck & Ruprecht, Göttingen, 1978, Studia Mathematica, Skript 4. MR 0463413 (573365)
23.: Richard B. Holmes, Geometric functional analysis and its applications, Springer-Verlag, New York, 1975, Graduate Texts in Mathematics, No. 24. MR 0410335 (53:14085)
24.: David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston Inc., Boston, MA, 1983, With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 0688651 (85h:14026)
25.: -, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston Inc., Boston, MA, 1984, Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 0742776 (86b:14017)
26.: Zeev Nehari, Conformal mapping, Dover Publications Inc., New York, 1975, Reprinting of the 1952 edition. MR 0377031 (51:13206)
27.: Vern Paulsen, Private communication.
28.: -, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867 (2004c:46118)
29.: Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow, 1986. MR 0868472 (88h:46111)
30.: Gilles Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), no. 2, 351-369. MR 1415321 (97f:47002)
31.: Donald Sarason, The $H\sp{p}$ spaces of an annulus, Mem. Amer. Math. Soc. No. 56 (1965), 78. MR 0188824 (32:6256)
32.: V. L. Vinnikov and S. I. Fedorov, On the Nevanlinna-Pick interpolation in multiply connected domains, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 254 (1998), no. Anal. Teor. Chisel i Teor. Funkts. 15, 5-27, 244. MR 1691394 (2000d:47031)

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
Email: sam@math.ufl.edu

DOI: 10.1090/S0894-0347-05-00491-1
PII: S 0894-0347(05)00491-1
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
Posted: 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
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1088-6834(e) ISSN: 0894-0347(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS