On trace-convex noncommutative polynomials

Klep, Igor; McCullough, Scott; Nelson, Chris

doi:10.1307/mmj/1457101814

Cited by 2 publications

(2 citation statements)

References 27 publications

(26 reference statements)

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“…Similarly, we say a function is trace convex if tr f A+B 2 ≤ tr f (A)+f (B) 2 . As happened in the case of monotonicity, a function f is trace convex if and only if f is convex [8,13]. In multivariable settings, the theory of joint trace convexity depends intensely on the expression being analyzed [9,2,3].…”

Section: Introductionmentioning

confidence: 99%

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Trace minmax functions and the radical Laguerre-Pólya class

Pascoe¹

2020

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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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Section: Introductionmentioning

confidence: 99%

“…In general, the current theory of tracial inequalities is real analytic and the theory of matrix inequalities is complex analytic. We give a class of trace functions that have nice complex analytic properties, which contrasts to existing literature [9,8,2,3,13].…”

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confidence: 96%