A Generalized Jentzsch Theorem

A Generalized Jentzsch Theorem Positivity (2005) 9:501–509 © Springer 2005 DOI 10.1007/s11117-004-8291-7 A.K. KITOVER Department of Mathematics, Community College of Philadelphia, 1700 Spring Garden Street Philadelphia, PA 19130 USA (e-mail: akitover@ccp.edu) Received 3 September 2004; accepted 13 December 2004 The Jentzsch theorem published in 1912 in [6] (See also [1, Theorem 9.44]) states the following. THEOREM 1. Let K : L [0, 1] → L [0, 1] be an integral operator defined 2 2 by Kx(t ) = K(s, t)x(s)ds, where K :[0, 1] × [0, 1] → R is a strictly positive continuous function. Then (1) The positive operator K is compact and its spectral radius r(K) is posi- tive. (2) The spectral radius r(K) is an eigenvalue of K and has a unique (up to a scalar multiple) strictly positive eigenfunction. (3) σ (K) ={r(K)}, where σ (K) is the peripheral spectrum of K. per per In the book by A.C. Zaanen [12, Theorem 137.3] the Jentzsch theo- rem was extended on the class of compact abstract integral operators (see e.g. [1, Definition 5.17]). DEFINITION 2. Let X be a vector lattice and X be the band of all order continuous functionals on X. An order bounded linear operator T : http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

A Generalized Jentzsch Theorem

Positivity , Volume 9 (3) – Dec 29, 2004
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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2005 by Springer
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-004-8291-7
Publisher site
See Article on Publisher Site

Abstract

Positivity (2005) 9:501–509 © Springer 2005 DOI 10.1007/s11117-004-8291-7 A.K. KITOVER Department of Mathematics, Community College of Philadelphia, 1700 Spring Garden Street Philadelphia, PA 19130 USA (e-mail: akitover@ccp.edu) Received 3 September 2004; accepted 13 December 2004 The Jentzsch theorem published in 1912 in [6] (See also [1, Theorem 9.44]) states the following. THEOREM 1. Let K : L [0, 1] → L [0, 1] be an integral operator defined 2 2 by Kx(t ) = K(s, t)x(s)ds, where K :[0, 1] × [0, 1] → R is a strictly positive continuous function. Then (1) The positive operator K is compact and its spectral radius r(K) is posi- tive. (2) The spectral radius r(K) is an eigenvalue of K and has a unique (up to a scalar multiple) strictly positive eigenfunction. (3) σ (K) ={r(K)}, where σ (K) is the peripheral spectrum of K. per per In the book by A.C. Zaanen [12, Theorem 137.3] the Jentzsch theo- rem was extended on the class of compact abstract integral operators (see e.g. [1, Definition 5.17]). DEFINITION 2. Let X be a vector lattice and X be the band of all order continuous functionals on X. An order bounded linear operator T :

Journal

PositivitySpringer Journals

Published: Dec 29, 2004

References

  • The peripheral spectrum of a nonnegative matrix
    McDonald, J.J.

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