Applications of Completions of Operator Matrices to Some Properties of Operator Products on Hilbert Spaces

Abstract

In this paper, we will investigate certain properties of some operator products on Hilbert spaces, by applications of completions of operator matrices. It is shown that, quite surprisingly, the invariance properties of the operator product $T_1T_2T_2^{(1,\ldots )}T_1^{(1,\ldots )}T_1T_2$ have a neat relationship with the properties of the reverse order laws for generalized inverses of the operator product $T_1T_2$. That is, the mixed-type reverse order laws

$$\begin{aligned} T_2\{1,\ldots \}T_1\{1,\ldots \}\subseteq (T_1T_2)\{1\} \end{aligned}$$

hold if and only if the operator product $T_1T_2T_2^{(1,\ldots )}T_1^{(1,\ldots )}T_1T_2$ is invariant, where $(1,\ldots )$ is taken respectively as (1), (1, 2), (1, 3), (1, 4), (1, 2, 3) as well as (1, 2, 4).