COSAM » Events » 2015 » September » Linear Algebra Seminar

 Linear Algebra Seminar Time: Sep 22, 2015 (04:00 PM) Location: Parker Hall 224 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.

Last updated: 09/17/2015