# Legendre–Fenchel transform of the spectral exponent of polynomials of weighted composition operators

Legendre–Fenchel transform of the spectral exponent of polynomials of weighted composition... For the spectral radius of weighted composition operators with positive weight e φ T α , $${\varphi\in C(X)}$$ , acting in the spaces L p (X, μ) the following variational principle holds $$\ln r(e^\varphi T_\alpha)=\max_{\nu\in M^1_\alpha} \left\{\int\limits_X\varphi d\nu-\frac{\tau_\alpha(\nu)}{p}\right\},$$ where X is a Hausdorff compact space, $${\alpha:X\mapsto X}$$ is a continuous mapping and τ α some convex and lower semicontinuous functional defined on the set $${M^1_\alpha}$$ of all Borel probability and α-invariant measures on X. In other words $${\frac{\tau_\alpha}{p}}$$ is the Legendre– Fenchel conjugate of ln r(e φ T α ). In this paper we consider the polynomials with positive coefficients of weighted composition operator of the form $${A_{\varphi, {\bf c}}= \sum_{k=0}^n e^{c_k} (e^{\varphi} T_{\alpha})^k}$$ , $${{\bf c}=(c_k)\in {\Bbb R}^{n+1}}$$ . We derive two formulas on the Legendre–Fenchel transform of the spectral exponent ln r(A φ,c ) considering it firstly depending on the function φ and the variable c and secondly depending only on the function φ, by fixing c. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Legendre–Fenchel transform of the spectral exponent of polynomials of weighted composition operators

, Volume 14 (3) – Jun 16, 2009
9 pages

/lp/springer_journal/legendre-fenchel-transform-of-the-spectral-exponent-of-polynomials-of-0HeE5u08n2
Publisher
Springer Journals
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-009-0023-6
Publisher site
See Article on Publisher Site

### Abstract

For the spectral radius of weighted composition operators with positive weight e φ T α , $${\varphi\in C(X)}$$ , acting in the spaces L p (X, μ) the following variational principle holds $$\ln r(e^\varphi T_\alpha)=\max_{\nu\in M^1_\alpha} \left\{\int\limits_X\varphi d\nu-\frac{\tau_\alpha(\nu)}{p}\right\},$$ where X is a Hausdorff compact space, $${\alpha:X\mapsto X}$$ is a continuous mapping and τ α some convex and lower semicontinuous functional defined on the set $${M^1_\alpha}$$ of all Borel probability and α-invariant measures on X. In other words $${\frac{\tau_\alpha}{p}}$$ is the Legendre– Fenchel conjugate of ln r(e φ T α ). In this paper we consider the polynomials with positive coefficients of weighted composition operator of the form $${A_{\varphi, {\bf c}}= \sum_{k=0}^n e^{c_k} (e^{\varphi} T_{\alpha})^k}$$ , $${{\bf c}=(c_k)\in {\Bbb R}^{n+1}}$$ . We derive two formulas on the Legendre–Fenchel transform of the spectral exponent ln r(A φ,c ) considering it firstly depending on the function φ and the variable c and secondly depending only on the function φ, by fixing c.

### Journal

PositivitySpringer Journals

Published: Jun 16, 2009

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