Spectral approximations of a normal operator

Author:
Richard Bouldin

Journal:
Proc. Amer. Math. Soc. **76** (1979), 279-284

MSC:
Primary 47B15

DOI:
https://doi.org/10.1090/S0002-9939-1979-0537088-2

MathSciNet review:
537088

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Abstract: If is a closed convex set in the complex plane then denotes all the normal (bounded linear) operators on the fixed separable Hilbert space *H* with spectrum contained in . The fixed operator *A* has *N* as an -approximant provided *N* belongs to and the operator norm equals , the distance from *A* to . With some hypothesis on , this note proves that the dimension of the convex set of all -approximants of normal operator *A* is where is the orthogonal complement of and is the unique distaince minimizing retract of the complex plane onto .

**[1]**R. H. Bouldin,*The convex structure of positive approximants for a given operator*, Acta Sci. Math. (Szeged)**37**(1975), 177-190. MR**0388152 (52:8989)****[2]**R. H. Bouldin and D. D. Rogers,*Normal dilations and operator approximations*, Acta Sci. Math. (Szeged)**39**(1977), 233-243. MR**0467362 (57:7221)****[3]**C. K. Chui, P. W. Smith and J. D. Ward,*Approximation with restricted spectra*, Math. Z.**144**(1975), 289-297; ibid.**146**(1976), 291-292. MR**0380465 (52:1365)****[4]**P. R. Halmos,*Some unsolved problems of unknown depth about operators on Hilbert space*, Proc. Roy. Soc. Edinburgh Sect. A**76A**(1976), 67-76. MR**0451002 (56:9292)****[5]**-,*Spectral approximants of normal operator*, Proc. Edinburgh Math. Soc.**19**(1974), 51-58. MR**0344935 (49:9674)****[6]**W. Rudin,*Real and complex analysis*, McGraw-Hill, New York, 1966. MR**0210528 (35:1420)****[7]**F. A. Valentine,*Convex sets*, McGraw-Hill, New York, 1964. MR**0170264 (30:503)**

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DOI:
https://doi.org/10.1090/S0002-9939-1979-0537088-2

Article copyright:
© Copyright 1979
American Mathematical Society