Details:
Speaker: Dr. Đinh Trung Hoà (Auburn)
Title: On characterization of operator monotone functions.
Abstract: The Loewner theorem states that for any number \(r\) between 0 and 1, for any positive matrices \(A, B,\)
$$
A \le B \mbox{ implies } A^r \le B^r.
$$
The above inequality fails, in general, when \(r>1\). We say that the function \(t^r\) (\(r \in [0, 1]\)) is operator monotone on \([0, \infty)\). Such kind of functions is important in quantum information theory.
The AGM inequality is well-known for positive numbers and still true for positive definite matrices. In this talk, we prove a reverse AGM inequality for matrices and show that this inequality characterizes operator monotone functions. More precisely, it will be shown that if for any positive definite matrices \(A, B\) the following inequality
$$
f((A+B)/2) \le f(A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2} A^{1/2} + 1/2 A^{1/2}|I-A^{-1/2}BA^{-1/2}|A^{1/2})
$$
holds true, then the function f is operator monotone on \([0, \infty)\).
Some related results and open questions also will be considered.
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