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

Abstract

For the spectral radius of weighted composition operators with positive weight e ϕ T α , ϕ ∈ C(X ), acting in the spaces L p (X, μ) the following variational principle holdswhere X is a Hausdorff compact space, α : X → X is a continuous mapping and τ α some convex and lower semicontinuous functional defined on the set M 1 α of all Borel probability and α-invariant measures on X . In other words τ α p is the LegendreFenchel 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 ϕ,c = n k=0 e c k (e ϕ T α ) k , c = (c k ) ∈ 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.

A variational formula on the Cramér function of series of independent random variables

“…Often generalizations of formulas from finite numbers parameters (variables) to the case of infinite ones are not obvious. Other examples of a generalizations of the convex conjugates of the logarithm of series of analytic functions, with applications to investigations of the convex conjugates of the spectral radius of the functions of weighted composition operators, one can find in [8,12].…”

Cramér transform and t-entropy

“…μ n for any n. Notice that the moment-generating function of X satisfies assumptions of Theorem 2.5 in [6] and for a > 0, by the formula (15), we obtain…”

“…and it is known that this power series possesses the convergence radius R not less than c. Moreover if the moment-generating function M X of a nonnegative random variable satisfies additionally condition lim t→R − M X (t) = +∞ then, by Theorem 2.5 in [6], we obtain the following formula on the convex conjugate of composition ln M X • exp depending on the moments of random variable X…”

“…2.3], the entropy function of infinite numbers of variables Consider a discrete random variable X taking values in N ∪ {0}; P(X = n) = p n . In this case it appears another opportunity of an application of Theorem 2.5 [6]. The probability-generating function of X has the form…”

“…It means that the effective domain D(λ * ) is contained in M 1 α . It turned out that considerations on the spectral exponent (the logarithm of the spectral radius) of some functions of WCO in the natural way lead us to investigate expressions which are similar to the cumulant generating functions of random variables (see [6]). Thus it appeared the natural idea to define operators which are moment generating functions of WCO and next to investigate their spectral exponent using tools related with the Cramér transform of given random variables.…”