Essentially subnormal operators

Author:
Nathan S. Feldman

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1171-1181

MSC (1991):
Primary 47B20; Secondary 47C15

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.

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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