, Volume 11, Issue 4, pp 549–562 | Cite as

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

  • Urszula Ostaszewska
  • Krzysztof Zajkowski


In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form
and acting in spaces L p (X, μ), where X is a compact topological space, φ k C(X), φ = (φ k ) k=1 N 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
$$\ln r(A_\varphi)=\max_{\nu\in{\mathfrak{Mes}}}\Big\{\nu(\varphi)-{\mathfrak{T}}(\nu)\Big\}.$$

Mathematics Subject Classification (2000)

47B37 47A10 47B65 


Positive operators spectral radius Fenchel-Legendre transform conditional expectation operators left inverses to composition operators 


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Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of BialystokBialystokPoland
  2. 2.Institute of MathematicsUniversity of BialystokBialystokPoland

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