Legendre-Fenchel Transform of the Spectral Exponent of Analytic Functions of Weighted Composition Operators

Abstract

<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be an analytic function with nonnegative coefficients. We derive the Legendre-Fenchel transform of the composition <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\ln \circ f \circ \exp</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ln</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo>∘</mml:mo> <mml:mi>f</mml:mi> <mml:mo>∘</mml:mo> <mml:mi>exp</mml:mi> <mml:mo>⁡</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> as a function depending on coefficients of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . We apply it to obtain the variational principle for the spectral exponent of operators that can be written as analytic functions of the weighted composition operators. </jats:p>