Compact operators whose real and imaginary parts are positive期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

Compact operators whose real and imaginary parts are positive

Authors:	Rajendra Bhatia Xingzhi Zhan

Institution:	Indian Statistical Institute, New Delhi 110 016, India ; Institute of Mathematics, Peking University, Beijing 100871, China

Abstract:	Let be a compact operator on a Hilbert space such that the operators $A = \frac{1}{2} (T + T^{})$ and $B = \frac{1}{2i}(T-T^{})$ are positive. Let $\{ s_{j}\}$ be the singular values of and $\{ \alpha _{j}\} , \{ \beta _{j}\}$ the eigenvalues of , all enumerated in decreasing order. We show that the sequence $\{ s^{2}_{j}\}$ is majorised by $\{ \alpha ^{2}_{j} + \beta ^{2}_{j}\}$ . An important consequence is that, when $p \ge 2, ~\Vert T\Vert ^{2}_{p}$ is less than or equal to $\Vert A\Vert ^{2}_{p} + \Vert B\Vert ^{2}_{p}$ , and when $1\le p \le 2,$ this inequality is reversed.

Keywords:	Compact operator positive operator singular values eigenvalues majorisation Schatten $p$-norms

	点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Proceedings of the American Mathematical Society》下载免费的PDF全文