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The representations of generalized inverses of lower triangular operators. (English) Zbl 1194.47004

For complex Hilbert spaces \(\mathcal H, \mathcal K\), let \(\mathcal B(\mathcal H, \mathcal K)\) denote the space of all bounded linear operators from \(\mathcal H\) into \(\mathcal K\). When \(\mathcal H = \mathcal K\), we denote \(\mathcal B(\mathcal H, \mathcal K)\) by \(\mathcal B(\mathcal H)\). For \(T \in \mathcal B (\mathcal H, \mathcal K)\), if there exists an operator \(T^{\dagger} \in \mathcal B(\mathcal K, \mathcal H)\) such that \(TT^{\dagger}T=T,~T^{\dagger}TT^{\dagger}=T^{\dagger},~TT^{\dagger} = (TT^{\dagger})^*,~ T^{\dagger}T=(T^{\dagger}T)^*\), then \(T^{\dagger}\) is called the Moore-Penrose inverse of \(T\). It is well-known that \(T\) has a Moore-Penrose inverse if and only if \(R(T)\), the range of \(T\), is closed and that, if it exists, then it is unique.
For \(A \in \mathcal B(\mathcal H), ~B \in \mathcal B(\mathcal K)\) and \(C \in \mathcal B(\mathcal H, \mathcal K)\), let \(M_{C}(A,B):\mathcal H \oplus \mathcal K \to\mathcal H \oplus \mathcal K\) be defined by \(M_{C}(A,B)=\left(\begin{smallmatrix} A & 0 \\ C & D \\ \end{smallmatrix}\right)\). Let \(M_{C}^{\dagger}(A,B)\) denote the Moore-Penrose inverse of \(M_{C}(A,B)\). Let \(R(A)=L_1\), \(N(A^*)=L_2\), \(N(P_{N(A)}C^*P_{N(B^*)}) \cap N(B^*) = L_3\), \(R(P_{N(B^*)}CP_{N(A)}) = L_4\) and \(R(B)=L_5\). Then \(\mathcal H \oplus \mathcal K\) has a direct-sum decomposition \(L_1 \oplus L_2 \oplus L_3 \oplus L_4 \oplus L_5\), considered here as the domain space (for \(M_{C}^{\dagger}(A,B)\)).
Also, let \(R(A^*)=M_1\), \(R(P_{N(A)}C^*P_{N(B^*)}) = M_2\), \(R(P_{N(B^*)}CP_{N(A)}) \cap N(A) = M_3\), \(N(B) = M_4\) and \(R(B^*)=M_5\). Then \(\mathcal H \oplus \mathcal K\) has another direct-sum decomposition \(M_1 \oplus M_2 \oplus M_3 \oplus M_4 \oplus M_5\), this time being considered as the codomain.
In the paper under review, the authors present sufficient conditions under which \(M_{C}^{\dagger}(A,B)\) exists and derive a \(5 \times 5\) block operator representation for the same, as an operator on \(\mathcal H \oplus \mathcal K\) with respect to the direct-sum decompositions of \(\mathcal H \oplus \mathcal K\), as mentioned above. As an application, an explicit representation of the Bott-Duffin inverse of a bounded linear operator on a Hilbert space is obtained.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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