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Trace minmax functions and the radical Laguerre-Polya class
Journal article   Open access   Peer reviewed

Trace minmax functions and the radical Laguerre-Polya class

J. E. Pascoe
Research in the mathematical sciences, v 8(1), 9
01 Mar 2021
DOI: https://doi.org/10.1007/s40687-021-00248-5
url
http://arxiv.org/abs/2008.05469View
SubmittedarXiv.org - Non-exclusive license to distribute,  Open

Abstract

Mathematics Physical Sciences Science & Technology
We classify functions f:(a, b)-> R which satisfy the inequality trf(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 Laguerre-Polya class. We derive an integral representation for such functions which is essentially a continuous version of the Hadamard factorization for functions in the Laguerre-Polya class. We apply our results to give some equivalent formulations of the Riemann hypothesis.

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