Double piling structure of matrix monotone functions and of matrix convex functions

✍ Scribed by Hiroyuki Osaka; Jun Tomiyama

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 132 KB
• Volume: 431
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2009.06.017

✦ Synopsis

There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix n-monotone functions and matrix n-convex functions, and we focus the following three assertions at each label n among them:

(i) f (0) 0 and f is n-convex in [0, α), (ii) For each matrix a with its spectrum in [0, α) and a contraction c in the matrix algebra M n ,

We show that for any n ∈ N two conditions (ii) and (iii) are equivalent. The assertion that f is n-convex with f (0) 0 implies that g(t) is (n -1)-monotone holds. The implication from (iii) to (i) does not hold even for n = 1. We also show in a limited case that the condition (i) implies (ii).