Inequivalent measures of noncompactness and the radius of the essential spectrum
Authors:
John MalletParet and Roger D. Nussbaum
Journal:
Proc. Amer. Math. Soc. 139 (2011), 917930
MSC (2010):
Primary 47H08, 46B20; Secondary 46B25, 46B45, 47A10, 47H10
Published electronically:
October 29, 2010
MathSciNet review:
2745644
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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 In general a map which assigns to each bounded set in a nonnegative real number and which shares most of the properties of is called a homogeneous measure of noncompactness or homogeneous MNC. Two homogeneous MNC's and on are called equivalent if there exist positive constants and with for all bounded sets . There are many results which prove the equivalence of various homogeneous MNC's. Working with where , we give the first examples of homogeneous MNC's which are not equivalent. 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 The formula is known to be true if is equivalent to , the Kuratowski MNC; however, as we show, it is in general false for MNC's which are not equivalent to . On the other hand, if denotes the unit ball in and is any homogeneous MNC, we prove that Our motivation for this study comes from questions concerning eigenvectors of linear and nonlinear conepreserving maps.
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Additional Information
John MalletParet
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:
http://dx.doi.org/10.1090/S000299392010105117
PII:
S 00029939(2010)105117
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 DMS0500674
The second author was partially supported by NSF Grant DMS0701171
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.
