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Trace functions as Laplace transforms. (English) Zbl 1111.47022

Summary: We study trace functions of the form \(t\mapsto \mathrm{Tr}f(A+tB)\) where \(f\) is a real function defined on the positive half-line and \(A\) and \(B\) are matrices such that \(A\) is positive definite and \(B\) is positive semidefinite. If \(f\) is non-negative and operator monotone decreasing, then such a trace function can be written as the Laplace transform of a positive measure. The question is related to the Bessis–Moussa–Villani conjecture [D.Bessis, P.Moussa and M.Villani, J. Math.Phys.16, No.11, 2318–2325 (1975; Zbl 0976.82501)].

MSC:

47A63 Linear operator inequalities
15A15 Determinants, permanents, traces, other special matrix functions
43A35 Positive definite functions on groups, semigroups, etc.

Citations:

Zbl 0976.82501
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References:

[1] Araki H., Ann. Sci. Ec. Normale Super. 6 pp 67– (1973)
[2] DOI: 10.1063/1.522463 · Zbl 0976.82501 · doi:10.1063/1.522463
[3] DOI: 10.1007/978-3-642-65755-9 · doi:10.1007/978-3-642-65755-9
[4] DOI: 10.1155/S0161171202011468 · Zbl 1001.15003 · doi:10.1155/S0161171202011468
[5] DOI: 10.1017/CBO9780511897191 · doi:10.1017/CBO9780511897191
[6] DOI: 10.1016/0024-3795(92)90268-F · Zbl 0745.15013 · doi:10.1016/0024-3795(92)90268-F
[7] DOI: 10.2977/prims/1195145324 · Zbl 0902.47013 · doi:10.2977/prims/1195145324
[8] DOI: 10.1016/S0024-3795(01)00392-5 · Zbl 1034.47005 · doi:10.1016/S0024-3795(01)00392-5
[9] Hansen F., J. Inequal. Pure Appl. Math. 5 pp 16– (2004)
[10] DOI: 10.2977/prims/1195164797 · Zbl 0829.46043 · doi:10.2977/prims/1195164797
[11] DOI: 10.1515/crll.1878.84.70 · doi:10.1515/crll.1878.84.70
[12] DOI: 10.1137/S0895479801387073 · Zbl 1007.68139 · doi:10.1137/S0895479801387073
[13] DOI: 10.1023/B:JOSS.0000019811.15510.27 · Zbl 1157.81313 · doi:10.1023/B:JOSS.0000019811.15510.27
[14] Nielsen H. P., Nyt Tidsskr. Mat. B 8 pp 86– (1897)
[15] DOI: 10.1007/978-3-642-50824-0 · doi:10.1007/978-3-642-50824-0
[16] DOI: 10.2977/prims/1195143229 · Zbl 0969.47010 · doi:10.2977/prims/1195143229
[17] Widom H., Lecture Notes in Mathematics 1573, in: Linear and Complex Analysis Problem Book 3, Part 1 pp 266– (1994)
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