Báo cáo hóa học: " Research Article Trace-Inequalities and Matrix-Convex Functions"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Trace-Inequalities and Matrix-Convex Functions | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 241908 12 pages doi 2010 241908 Research Article Trace-Inequalities and Matrix-Convex Functions Tsuyoshi Ando Hokkaido University Emeritus Shiroishi-ku Hongo-dori 9 Minami 4-10-805 Sapporo 003-0024 Japan Correspondence should be addressed to Tsuyoshi Ando ando@ Received 8 October 2009 Accepted 30 November 2009 Academic Editor Anthony To Ming Lau Copyright 2010 Tsuyoshi Ando. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. A real-valued continuous function f t on an interval a p gives rise to a map X f X via functional calculus from the convex set of n X n Hermitian matrices all of whose eigenvalues belong to the interval. Since the subpace of Hermitian matrices is provided with the order structure induced by the cone of positive semidefinite matrices one can consider convexity of this map. We will characterize its convexity by the following trace-inequalities Tr f B - f A C - B Tr f C - f B B - A for A B C. A related topic will be also discussed. 1. Introduction and Theorems Let f t be a real-valued continuous function defined on an open interval a p of the real line. The function f t is said to be convex if f Xa 1 - X b Xf a 1 - X f b 0 X 1 a a b p . We referee to 1 for convex functions. Under continuity the requirement can be restricted only to the case X 1 2 that is f a b f Ò f b a a b p . It is well known that when f t is a C1-function its convexity is characterized by the condition on the derivative f b - f b - rn b a b - t b n 2 Fixed Point Theory and Applications and further when f t is a C2-function by the condition on the second derivative f V 0 a b p . On the other hand it is easy to see that is equivalent to the following requirement on the divided difference

Không thể tạo bản xem trước, hãy bấm tải xuống
Đã phát hiện trình chặn quảng cáo AdBlock
Trang web này phụ thuộc vào doanh thu từ số lần hiển thị quảng cáo để tồn tại. Vui lòng tắt trình chặn quảng cáo của bạn hoặc tạm dừng tính năng chặn quảng cáo cho trang web này.