Essentially subnormal operators

Feldman, Nathan

doi:10.1090/S0002-9939-99-05053-4

Essentially subnormal operators

Author: Nathan S. Feldman
Journal: Proc. Amer. Math. Soc. 127 (1999), 1171-1181
MSC (1991): Primary 47B20; Secondary 47C15
DOI: https://doi.org/10.1090/S0002-9939-99-05053-4
MathSciNet review: 1625741
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Abstract: An operator is essentially subnormal if its image in the Calkin algebra is subnormal. We shall characterize the essentially subnormal operators as those operators with an essentially normal extension. In fact, it is shown that an essentially subnormal operator has an extension of the form ``normal plus compact''.

The essential normal spectrum is defined and is used to characterize the essential isometries. It is shown that every essentially subnormal operator may be decomposed as the direct sum of a subnormal operator and some irreducible essentially subnormal operators. An essential version of Putnam's Inequality is proven for these operators. Also, it is shown that essential normality is a similarity invariant within the class of essentially subnormal operators. The class of essentially hyponormal operators is also briefly discussed and several examples of essentially subnormal operators are given.

1. A. Aleman, Subnormal Operators with Compact Self-commutator, Manuscripta Math. 91, No. 3 (1996), 353-367. MR 97j:47036
2. C. Apostol, H. Bercovici, C. Foias and C. Pearcy, Invariant subspaces, dilation theory and the structure of the predual of a dual algebra, J. Funct. Anal. 63 (1985), 369-404. MR 87i:470049
3. S. Axler and J. Shapiro, Putnam's Theorem, Alexander's spectral area estimate, and VMO, Math. Ann. 271 (1985), 161-183. MR 87b:30053
4. J. Bram, Subnormal Operators, Duke Math. J. 22 (1955), 75-94. MR 16:835a
5. L.G. Brown, R.G. Douglas and P.A. Fillmore, Unitary equivalence modulo the compact operators and extensions of -algebras, Lect. Notes in Math. 345, Springer-Verlag, (1973), 58-128. MR 52:1378
6. J. W. Bunce and J.A. Deddens, On the normal spectrum of a subnormal operator, Proc. Amer. Math. Soc. 63 (1977), 107-110. MR 55:8863
7. J.B. Conway, The Theory of Subnormal Operators, Amer. Math. Soc., Providence, RI, 1991. MR 92h:47026
8. J.B. Conway and N.S. Feldman, The Essential Selfcommutator of a Subnormal Operator, Proc. Amer. Math. Soc. 125, No. 1 (1997), 243-244. MR 97c:47022
9. J.B. Conway and C.R. Putnam, An irreducible subnormal operator with infinite multiplicities, J. Operator Theory 13 (1985), 291-297. MR 86b:47040
10. K.R. Davidson, -algebras by Example, Amer. Math. Soc., Providence, RI, 1996. MR 97i:46095
11. R. Lange, Essentially Subnormal Operators and K-Spectral Sets, Proc. Amer. Math. Soc. 88 (1983), 449-453. MR 84e:47008
12. H. Hedenmalm, S. Richter, and K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math. 477 (1996), 13-30. MR 97i:46044
13. H. Radjavi and P. Rosenthal, On roots of normal operators, J. Math. Anal. Appl. 34 (1971), 653-664. MR 43:3829
14. J.G. Stampfli and B.L. Wadhwa, An asymmetric Putnam-Fuglede theorem for dominant operators, Indiana Univ. Math. J. 25 (1976), 359-365. MR 53:14197
15. L.R. Williams, Restrictions of Essentially Normal Operators, Rocky Mtn J. of Math. 20 No. 2 (1990), 613-618. MR 91g:47015

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Additional Information

Nathan S. Feldman
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996–1300
Address at time of publication: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: feldman@math.utk.edu, feldman@math.msu.edu

DOI: https://doi.org/10.1090/S0002-9939-99-05053-4
Keywords: Essentially subnormal operator, essentially normal operator
Received by editor(s): August 1, 1997
Additional Notes: This paper was written while the author was a graduate student working under the direction of Professor John B. Conway. He received partial support from the NSF grant DMS–9401027.
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society