Anti-Loewner matrices : Numerical radius and unitarity (Structural study of operators via spectra or numerical ranges)
Chikara Hidaka, Takashi Sano · Institutional Repositories DataBase (IRDB) · 2012
We review results on two topics by Hidaka and Sano; Sano and A. Uchiyama.For details, we refer [4,5]. 1 Anti-Loewner matrices Let $f$ be a positive $C^{1}$ function on $(0, \infty)$ .Let $H^{n}$ be the subspace of $\mathbb{C}^{n}$ consisting of all $x=(x_{1}, \ldots, x_{n})^{T}\in \mathbb{C}^{n}$ for which $\sum_{i=1}^{n}x_{i}=0$ .An $n\cross n$ Hermitian matrix $A$ is said to be conditionally positive definite (c.p. $d$ .for short) if $\langle x,$ $Ax\rangle\geqq 0$ for all $x\in H^{n}$ , and conditionally negative definite (c.n.$d$ .for short) $if-A$ is c.p. $d$ .For positive numbers $t_{1},$ $\ldots,$ $t_{n}$ , the matrices $K_{f}(t_{1}, \ldots, t_{n})=[\frac{f(t_{i})+f(t_{j})}{t_{i}+t_{j}}]$ have been of some interest.We call it an anti-Loewner matrix.Kwong showed that if $f$ is a non-negative operator monotone function on $(0, \infty)$ then all $K_{f}$ are p.s.Recently, Audenaert in [2] gives a characterisation of functions $f$ for which all $K_{f}$ are p.s. $d$ ; by [2, Theorem 2.1], for a positive $C^{1}$ function $f$ on $(0, \infty)$ , all $K_{f}$ are p.s. $d$ .if and only if $f(\sqrt{t})\sqrt{t}$ is matrix monotone of any order $n$ , i.e., operator monotone.Hence, such a function $f$ is of the form $f(t)= \frac{\alpha}{t}+\beta t+\int_{0}^{\infty}\frac{t}{\lambda+t^{2}}d u(\lambda)$ , (1.1)