Essentially subnormal operators and -spectral sets

Author:
Ridgley Lange

Journal:
Proc. Amer. Math. Soc. **88** (1983), 449-453

MSC:
Primary 47A15; Secondary 47B20

DOI:
https://doi.org/10.1090/S0002-9939-1983-0699412-0

MathSciNet review:
699412

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an essentially subnormal operator. We give six conditions which are equivalent to the spectrum of being a -spectral set. From this follow two corollaries which give sufficient conditions for invariant subspaces of essentially subnormal operators. Several examples are given that show that some essentially subnormal operators are not essentially normal nor perturbations of subnormal operators.

**[1]**L. G. Brown, R. G. Douglas and P. A. Fillmore,*Unitary equivalence modulo the compact operators and extensions of**-algebras*, Proc. Conf. Operator Theory, Lecture Notes in Math., vol. 345, Springer, 1973. MR**0380478 (52:1378)****[2]**J. W. Calkin,*Two-sided ideals and congruences in the ring of bounded operators in Hilbert space*, Ann. of Math. (2)**42**(1941), 839-873. MR**0005790 (3:208c)****[3]**K. R. Davidson and C.-K. Fong,*An algebra which is not closed in the Calkin algebra*, Pacific J. Math.**72**(1977), 57-58. MR**0463931 (57:3869)****[4]**P. A. Fillmore, J. G. Stampfli and J. P. Williams,*On the essential numerical range, the essential spectrum, and a problem of Halmos*, Acta Sci. Math. (Szeged)**33**(1972), 179-192. MR**0322534 (48:896)****[5]**R. F. Olin,*Functional relationships between a subnormal operator and its minimal normal extension*, Pacific J. Math**63**(1976), 221-229. MR**0420324 (54:8338)****[6]**J. G. Stampfli,*Compact perturbations, normal eigenvalues, and a problem of Salinas*, J. London Math. Soc.**9**(1974), 165-175. MR**0365196 (51:1449)****[7]**-,*An extension of Scott Brown's subspace theorem:**-spectral sets*, J. Operator Theory**3**(1980), 3-21.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47A15,
47B20

Retrieve articles in all journals with MSC: 47A15, 47B20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0699412-0

Keywords:
Essentially subnormal operators,
-spectral set,
invariant subspace

Article copyright:
© Copyright 1983
American Mathematical Society