## The failure of rational dilation on a triply connected domain

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- by Michael A. Dritschel and Scott McCullough PDF
- J. Amer. Math. Soc.
**18**(2005), 873-918 Request permission

## 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.## References

- 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**397468**, DOI 10.1016/0001-8708(76)90023-2 - Jim Agler,
*Rational dilation on an annulus*, Ann. of Math. (2)**121**(1985), no. 3, 537–563. MR**794373**, DOI 10.2307/1971209 - Jim Agler,
*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**
AH Jim Agler and John Harland, - Jim Agler and John E. McCarthy,
*Nevanlinna-Pick interpolation on the bidisk*, J. Reine Angew. Math.**506**(1999), 191–204. MR**1665697**, DOI 10.1515/crll.1999.004 - Jim Agler and John E. McCarthy,
*Pick interpolation and Hilbert function spaces*, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR**1882259**, DOI 10.1090/gsm/044 - William Arveson,
*Subalgebras of $C^{\ast }$-algebras. II*, Acta Math.**128**(1972), no. 3-4, 271–308. MR**394232**, DOI 10.1007/BF02392166 - William B. Arveson,
*Subalgebras of $C^{\ast }$-algebras*, Acta Math.**123**(1969), 141–224. MR**253059**, DOI 10.1007/BF02392388 - 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**, DOI 10.1007/BF01192041 - 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**, DOI 10.1006/jfan.1998.3278 - 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** - 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** - Stefan Bergman and Bruce Chalmers,
*A procedure for conformal mapping of triply-connected domains*, Math. Comp.**21**(1967), 527–542. MR**228663**, DOI 10.1090/S0025-5718-1967-0228663-3 - 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** - Kevin F. Clancey,
*Representing measures on multiply connected planar domains*, Illinois J. Math.**35**(1991), no. 2, 286–311. MR**1091446** - John B. Conway,
*Functions of one complex variable*, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR**503901**, DOI 10.1007/978-1-4612-6313-5 - John B. Conway,
*The theory of subnormal operators*, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR**1112128**, DOI 10.1090/surv/036 - H. M. Farkas and I. Kra,
*Riemann surfaces*, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR**1139765**, DOI 10.1007/978-1-4612-2034-3 - John D. Fay,
*Theta functions on Riemann surfaces*, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR**0335789** - Stephen D. Fisher,
*Function theory on planar domains*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A second course in complex analysis; A Wiley-Interscience Publication. MR**694693** - Helmut Grunsky,
*Lectures on theory of functions in multiply connected domains*, Studia Mathematica: Skript, vol. 4, Vandenhoeck & Ruprecht, Göttingen, 1978. MR**0463413** - Richard B. Holmes,
*Geometric functional analysis and its applications*, Graduate Texts in Mathematics, No. 24, Springer-Verlag, New York-Heidelberg, 1975. MR**0410335**, DOI 10.1007/978-1-4684-9369-6 - 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**688651**, DOI 10.1007/978-1-4899-2843-6 - David Mumford,
*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**742776**, DOI 10.1007/978-0-8176-4578-6 - Zeev Nehari,
*Conformal mapping*, Dover Publications, Inc., New York, 1975. Reprinting of the 1952 edition. MR**0377031**
VIP Vern Paulsen, - Vern Paulsen,
*Completely bounded maps and operator algebras*, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR**1976867** - Vern I. Paulsen,
*Completely bounded maps and dilations*, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR**868472** - 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**, DOI 10.1090/S0894-0347-97-00227-0 - Donald Sarason,
*The $H^{p}$ spaces of an annulus*, Mem. Amer. Math. Soc.**56**(1965), 78. MR**188824** - 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 (Russian, with Russian summary); English transl., J. Math. Sci. (New York)**105**(2001), no. 4, 2109–2126. MR**1691394**, DOI 10.1023/A:1011368822699

*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*.

## 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
- 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 - © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2163865