## Rota’s theorem for general functional Hilbert spaces

HTML articles powered by AMS MathViewer

- by Joseph A. Ball PDF
- Proc. Amer. Math. Soc.
**64**(1977), 55-61 Request permission

## Abstract:

By a theorem of G.-C. Rota, every (linear) operator*T*on a Hilbert space with spectral radius less than one is similar to the adjoint of the unilateral shift

*S*of infinite multiplicity restricted to an invariant subspace. This theorem is shown to be true in a rather general context, where

*S*is multiplication by

*z*on a Hilbert space of functions analytic on an open subset

*D*of the complex plane, and

*T*is an operator with spectrum contained in

*D*. A several-variable version for an

*N*-tuple of commuting operators with a corollary concerning complete spectral sets is also presented.

## 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 - William Arveson,
*Subalgebras of $C^{\ast }$-algebras. II*, Acta Math.**128**(1972), no. 3-4, 271–308. MR**394232**, DOI 10.1007/BF02392166 - C. A. Berger and B. I. Shaw,
*Intertwining, analytic structure, and the trace norm estimate*, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973, pp. 1–6. MR**0361885** - Stefan Bergman,
*The Kernel Function and Conformal Mapping*, Mathematical Surveys, No. 5, American Mathematical Society, New York, N. Y., 1950. MR**0038439**, DOI 10.1090/surv/005 - Joseph Bram,
*Subnormal operators*, Duke Math. J.**22**(1955), 75–94. MR**68129** - James E. Brennan,
*Invariant subspaces and rational approximation*, J. Functional Analysis**7**(1971), 285–310. MR**0423059**, DOI 10.1016/0022-1236(71)90036-x - Douglas N. Clark,
*On commuting contractions*, J. Math. Anal. Appl.**32**(1970), 590–596. MR**267407**, DOI 10.1016/0022-247X(70)90281-7 - R. G. Douglas and Carl Pearcy,
*Invariant subspaces of non-quasitriangular operators*, Proc. Conf. Operator Theory (Dalhousie Univ., Halifax, N.S., 1973), Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973, pp. 13–57. MR**0358391** - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - Domingo A. Herrero,
*A Rota universal model for operators with multiply connected spectrum*, Rev. Roumaine Math. Pures Appl.**21**(1976), no. 1, 15–23. MR**407628** - Gian-Carlo Rota,
*On models for linear operators*, Comm. Pure Appl. Math.**13**(1960), 469–472. MR**112040**, DOI 10.1002/cpa.3160130309 - Donald Sarason,
*The $H^{p}$ spaces of an annulus*, Mem. Amer. Math. Soc.**56**(1965), 78. MR**188824** - Allen L. Shields,
*Weighted shift operators and analytic function theory*, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR**0361899** - Dan Voiculescu,
*Norm-limits of algebraic operators*, Rev. Roumaine Math. Pures Appl.**19**(1974), 371–378. MR**343082**

## Additional Information

- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**64**(1977), 55-61 - MSC: Primary 47A45; Secondary 47A25
- DOI: https://doi.org/10.1090/S0002-9939-1977-0461176-0
- MathSciNet review: 0461176