Operator matrix representations of the Generalised Bott–Duffin Inverse

Abstract

In this paper, we give the operator matrix representation of the generalised Bott–Duffin inverse $$A_{(\mathcal {M},\mathcal {N})}^{\dagger }= P_{\mathcal {M},\mathcal {N}} \left( AP_{\mathcal {M},\mathcal {N}} + P_{\mathcal {N},\mathcal {M}}\right) ^\dagger $$ using results that concern lower-triangular operator matrices. Specially, we extend some previous results and we give necessary and sufficient conditions for $$A_{(\mathcal {M},\mathcal {N})}^\dagger = (P_{\mathcal {M},\mathcal {N}}AP_{\mathcal {M},\mathcal {N}})^\dagger $$ to hold.