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Invariance properties of an operator product involving generalized inverses. (English) Zbl 1229.47003
For Hilbert spaces $\mathbb{H}$ and $\mathbb{K}$, let $L (\mathbb{H},\mathbb{K})$ denote the space of all bounded linear operators from $\mathbb{H}$ into $\mathbb{K}$. The Moore-Penrose (generalized) inverse of $T \in L (\mathbb{H},\mathbb{K})$ (if it exists) is the unique operator $T^{\dagger} \in L (\mathbb{K},\mathbb{H})$ satisfying the following equations known as Penrose equations: $TXT=T$; $XTX=X$; $(TX)^*=TX$; $(XT)^*=XT$. It is well known that $T^{\dagger}$ exists if and only if the range space of $T$, $R(T)$, is closed. Let $T\{1\}$ denote the set of all bounded operators $X$ which satisfy the equation $TXT=T$, and denote an arbitrary element in $T\{1\}$ by $T^{(1)}$. Similar subclasses of generalized inverses are defined analogously. Let $T_1 \in L(\mathbb{L}, \mathbb{H})$, $T_2 \in L(\mathbb{L}, \mathbb{K})$ and $T_3 \in L(\mathbb{H}, \mathbb{K})$. The objective of the present article is to study conditions under which the operator product $T_1T^{(1)}_2T_3$ is independent of the choice of $T^{(1)} \in T\{1\}$ (and also other subclasses of generalized inverses). We provide a sample result: $T_1T^{(1)}_2T_3$ is independent of the choice of $T^{(1)} \in T\{1\}$ if and only if $R(T_1^*) \subseteq R(T_2^*)$ and $R(T_3) \subseteq R(T_2)$.

MSC:
47A05General theory of linear operators
15A09Matrix inversion, generalized inverses
15A24Matrix equations and identities
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