The inverse of operator matrix A where A ≥ I

Abstract EN

Let H and K be Hilbert spaces and let H ⊕ K be the cartesian product of them.

Let B (H), B (K), B (H ⊕ K), B (K, H), B (H, K) be the Banach spaces of bounded (continuous) operators on H, K, H ⊕ K, and from K into H and from H into K respectively.

In this paper we find the inverse of operator matrix A = ⌊ (B@D) (C@E)⌋ ∈B (H ⊕ K) where B ∈ B (H), C ∈ B (K, H), D ∈ B (H, K), E ∈ B (K) and A ≥ H K I ⊕ where H K I ⊕ is the identity operator on H ⊕ K.