NORM AND ANTI-NORM INEQUALITIES FOR POSITIVE SEMI-DEFINITE MATRICES

作者：Bourin Jean Christophe^*; Hiai Fumio

来源：International Journal of Mathematics, 2011, 22(8): 1121-1138.

DOI：10.1142/S0129167X1100715X

• Matrix
• operator
• trace
• symmetric norm
• symmetric anti-norm
• convex function
• concave function
• subadditivity
• superadditivity
• majorization

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if g(t) = Sigma(m)(k-0) a(k)t(k) is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms,
parallel to g(A + B)parallel to(1/m) <= parallel to g(A)parallel to(1/m) + parallel to g(B)parallel to(1/m).
To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten q-norms for q is an element of (0, 1] and q < 0. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let f : [0, infinity) -> [0, infinity) be concave and p is an element of (1, infinity). If f(p)(t) is superadditive, then Tr f(A) >= (Sigma(m)(i= 1) f(p)(a(ii)))(1/p) for all positive m x m matrix A = [a(ij)]. Furthermore, for the normalized trace tau, we consider functions phi(t) and f(t) for which the functional A bar right arrow phi omicron tau omicron f(A) is convex or concave, and obtain a simple analytic criterion.

• 出版日期2011-8