# Inequalities for the spectral radius of non-negative functions

Inequalities for the spectral radius of non-negative functions Let (X, μ) be a σ-finite measure space and M(X × X)+ a cone of all equivalence classes of (almost everywhere equal) non-negative measurable functions on a product measure space X × X. Using Luxemburg-Gribanov theorem we define a product * on M(X × X)+. Given a function seminorm h : M(X × X)+ → [0, ∞], we introduce the spectral radius r h (f) of f ∈ M(X × X)+ with respect to h and *. We give several examples. In particular, r h (f) provides a generalization and unification of the spectral radius and its max version, the maximum cycle geometric mean, of a non-negative matrix. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Inequalities for the spectral radius of non-negative functions

, Volume 13 (1) – Apr 5, 2008
18 pages

Publisher
Springer Journals
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/s11117-008-2188-9
Publisher site
See Article on Publisher Site

### Abstract

Let (X, μ) be a σ-finite measure space and M(X × X)+ a cone of all equivalence classes of (almost everywhere equal) non-negative measurable functions on a product measure space X × X. Using Luxemburg-Gribanov theorem we define a product * on M(X × X)+. Given a function seminorm h : M(X × X)+ → [0, ∞], we introduce the spectral radius r h (f) of f ∈ M(X × X)+ with respect to h and *. We give several examples. In particular, r h (f) provides a generalization and unification of the spectral radius and its max version, the maximum cycle geometric mean, of a non-negative matrix.

### Journal

PositivitySpringer Journals

Published: Apr 5, 2008

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