A Stieltjes representation of order $p$ for $ \mathrm{tr}[(A+tB)^{-p}]$: Heinävaara's measure, B-splines, and Stahl's theorem

Abstract

For $A$ positive definite, $B$ positive semidefinite and $p\ge1$ we show that $\mathrm{tr}[(A+tB)^{-p}]=\int(1+t\ell)^{-p}\,dρ_p(\ell)$ for a positive measure $ρ_p$ on the spectral interval of $L=A^{-1/2}BA^{-1/2}$, and we identify the measure: it has atoms $\mathrm{tr}[(P_sA^{-1}P_s)^p]$ at the eigenvalues of $L$ and a continuous density, an integral of Heinävaara's tracial joint spectral measure along rays. For integer $p$ the positivity of $ρ_p$ follows from the positivity of Heinävaara's measure alone, and the density is a sum of Curry-Schoenberg B-splines; for real $p$ the representation is obtained by inverting Heinävaara's transform. For $0<p<1$ the function is a Stieltjes function of order $p$ only if $[A,B]=0$, but a positive representation of order one holds, with an explicit density. Along $A_p=pI+A_0$, $p\to\infty$, the measures converge to the measure of Stahl's theorem (the Bessis-Moussa-Villani conjecture) for $\mathrm{tr}\,e^{-(A_0+tB)}$.