Journal article
Trace minmax functions and the radical Laguerre-Polya class
Research in the mathematical sciences, v 8(1), 9
01 Mar 2021
Abstract
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.
Metrics
Details
- Title
- Trace minmax functions and the radical Laguerre-Polya class
- Creators
- J. E. Pascoe - University of Florida
- Publication Details
- Research in the mathematical sciences, v 8(1), 9
- Publisher
- SPRINGER INT PUBL AG
- Number of pages
- 13
- Grant note
- DMS-1953963 / NSF Analysis Grant
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000613471600001
- Scopus ID
- 2-s2.0-85100079471
- Other Identifier
- 991021879625004721
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- Web of Science research areas
- Mathematics