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

Spectral Exponent of Finite Sums of Weighted Positive Operators in L p -Spaces 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 ) 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 $$\ln r(A_\varphi)=\max_{\nu\in{\mathfrak{Mes}}}\Big\{\nu(\varphi)-{\mathfrak{T}}(\nu)\Big\}.$$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 11 (4) – Sep 26, 2007
14 pages

/lp/springer_journal/spectral-exponent-of-finite-sums-of-weighted-positive-operators-in-l-p-dXTVdmoWpa
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/s11117-007-2096-4
Publisher site
See Article on Publisher Site

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

### Journal

PositivitySpringer Journals

Published: Sep 26, 2007

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations