Spectral Exponent of Finite Sums of Weighted Positive Operators in L^p-Spaces

Abstract

In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form

$$A_\varphi=\sum_{k=1}^Ne^{\varphi_k}U_k$$

and acting in spaces L^p(X, μ), where X is a compact topological space, φ _k ∈C(X), φ = (φ _k ) ^N_k=1 ∈C(X)^N, and $U_k:L^p(X,\mu)\mapsto L^p(X,\mu)$ are linear positive operators (U _k f≥ 0 for f≥ 0). We consider the spectral exponent ln r(A _φ ) as a functional depending on vector-function φ. We prove that ln r(A _φ ) is continuous and on a certain subspace ${\mathfrak{C}}(X)^N$ of C(X)^N is also convex. This yields that the spectral exponent is the Fenchel-Legendre transform of a convex functional ${\mathfrak{T}}$ defined on a set ${\mathfrak{Mes}}$ of continuous linear positive and normalized functionals on the subspace ${\mathfrak{C}}(X)^N$ of coefficients φ that is

$$\ln r(A_\varphi)=\max_{\nu\in{\mathfrak{Mes}}}\Big\{\nu(\varphi)-{\mathfrak{T}}(\nu)\Big\}.$$