Inequivalent measures of noncompactness and the radius of the essential spectrum

Authors:
John Mallet-Paret and Roger D. Nussbaum

Journal:
Proc. Amer. Math. Soc. **139** (2011), 917-930

MSC (2010):
Primary 47H08, 46B20; Secondary 46B25, 46B45, 47A10, 47H10

DOI:
https://doi.org/10.1090/S0002-9939-2010-10511-7

Published electronically:
October 29, 2010

MathSciNet review:
2745644

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Abstract | References | Similar Articles | Additional Information

Abstract: The Kuratowski measure of noncompactness on an infinite dimensional Banach space assigns to each bounded set in a nonnegative real number by the formula

Further, if is any complex, infinite dimensional Banach space and is a bounded linear map, one can define , where denotes the essential spectrum of . One can also define

Our motivation for this study comes from questions concerning eigenvectors of linear and nonlinear cone-preserving maps.

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Additional Information

**John Mallet-Paret**

Affiliation:
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Email:
jmp@dam.brown.edu

**Roger D. Nussbaum**

Affiliation:
Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Email:
nussbaum@math.rutgers.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10511-7

Keywords:
Measure of noncompactness,
essential spectral radius,
cone map.

Received by editor(s):
September 21, 2009

Received by editor(s) in revised form:
January 16, 2010

Published electronically:
October 29, 2010

Additional Notes:
The first author was partially supported by NSF Grant DMS-0500674

The second author was partially supported by NSF Grant DMS-0701171

Communicated by:
Nigel J. Kalton

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.