Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkoff-Kellogg theorem

Raffaele Chiappinelli

doi:10.3934/proc.2007.2007.260

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2007, Volume 2007: 260-268. Doi: 10.3934/proc.2007.2007.260 open access

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Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkoff-Kellogg theorem

Raffaele Chiappinelli^1,

1.
Dipartimento di Scienze Matematiche ed Informatiche, Pian dei Mantellini 44, 53100 Siena

Received: June 30, 2006

Revised: March 31, 2007

Published: August 2007

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Abstract

It is well known that any (nontrivial) linear compact self-adjoint operator acting in a Hilbert space possesses at least one non-zero eigenvalue. We present a generalization of this to nonlinear mappings as in the title, and discuss the relations of our results with the Birkhoff-Kellogg Theorem on one side, and with the spectral properties of self-adjoint operators on the other.

Keywords:

self-adjoint operators,
Palais-Smale condition.

Mathematics Subject Classification: Primary: 47J10; Secondary: 47J10.

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Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkoff-Kellogg theorem

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