×

Some characterizations of self-adjoint operators. (English) Zbl 0572.47011

Let B(H) denote the algebra of all bounded linear operators on a separable Hilbert space H. The following two theorems are characterizations of self-adjoint and positive operators and were obtained by C. K. Fong and V. I. Istrǎtescu [Proc. Am. Math. Soc. 76, 107-112 (1979; Zbl 0436.47015)] and C. K. Fong and S. K. Tsui [J. Oper. Theory. 5, 73-76 (1981; Zbl 0489.47009)] respectively.
Theorem 1. An operator \(T\in B(H)\) is self-adjoint if and only if \(| T|^ 2\leq (ReT)^ 2.\)
Theorem 2. An operator \(T\in B(H)\) is positive if and only if \(| T| \leq ReT.\)
The purpose of this paper is to generalize Theorem 1 as well as to present a new proof of Theorem 2. A characterization modulo \(C_ p\) (the Schatten p-class) of self-adjoint operators is also given.

MSC:

47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B20 Subnormal operators, hyponormal operators, etc.
PDFBibTeX XMLCite