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.
We generalize some known results for the Hadamard weighted geometric mean of positive kernel operators and apply them to obtain some inequalities concerning the generalized spectral radius and its max version. We also give a new description of the maximum cycle geometric mean.
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Peperko, A. Inequalities for the spectral radius of non-negative functions. Positivity 13, 255–272 (2009). https://doi.org/10.1007/s11117-008-2188-9
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DOI: https://doi.org/10.1007/s11117-008-2188-9