Trace minmax functions and the radical Laguerre-Pólya class

2020/08/12 by Pascoe, J. E.
#11M26 #46E22 #46L52 Secondary 32A70 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT) #Primary 46L54

paper · doi:10.48550/arxiv.2008.05469

Abstract

We classify functions f:(a,b)→ ℝ which satisfy the inequality tr f(A)+f(C)≥ tr f(B)+f(D) when A≤ B≤ C are self-adjoint matrices, D= A+C-B, the so-called trace minmax functions. (Here A≤ B if B-A is positive semidefinite, and f is evaluated via the functional calculus.) A function is trace minmax if and only if its derivative analytically continues to a self map of the upper half plane. The negative exponential of a trace minmax function g=e^-f satisfies the inequality det g(A) det g(C)≤ det g(B) det g(D) for A, B, C, D as above. We call such functions determinant isoperimetric. We show that determinant isoperimetric functions are in the "radical" of the the Laguerre-Pólya class. We derive an integral representation for such functions which is essentially a continuous version of the Hadamard factorization for functions in the the Laguerre-Pólya class. We apply our results to give some equivalent formulations of the Riemann hypothesis.

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