Department Mathematik



Mathematisches Kolloquium

Am Mittwoch, 27. Juli 2005, um 16 Uhr c.t. spricht

Prof. Dr. Barry Simon
(California Institute of Technology)

im Hörsaal E27 über das Thema

The lost proof of Loewner's Theorem

Zusammenfassung: A real-valued function, F, on an interval (a,b) is called matrix monotone if F(A) < F(B) whenever A and B are finite matrices of the same order with eigenvalues in (a, b) and A < B. In 1934, Loewner proved the remarkable theorem that F is matrix monotone if and only if F is real analytic with continuations to the upper and lower half planes so that Im F > 0 in the upper half plane. This deep theorem has evoked enormous interest over the years and a number of alternate proofs. There is a lovely 1954 proof that seems to have been "lost" in the that the proof is not mentioned in various books and review article presentations of the subject, and I have found no reference to the proof since 1960. The proof uses continued fractions. I'll provide background on the subject and then discuss the lost proof and a variant of that proof which I've found, which even avoids the need for estimates, and proves a stronger theorem.
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