Extension of the result by Han J. K., Lee H. Y. and Lee W. Y

Dragan S. Djordjević, Nikola Sarajlija

Published 2022-02-10Version 1

We extend well known result of Han J. K., Lee H. Y. and Lee W. Y. \cite[Theorem 5.2]{HAN} related to invertibility of operator matrices to case of $3\times3$ upper triangular operator matrices. We provide necessary and sufficient conditions for invertibility in the framework of arbitrary Banach spaces, and explain under which additional assumption these conditions become equivalent. At the end, we construct an illustrating example to show that without additional assumption equivalence does not hold even in separable Hilbert spaces. Our main tools are ghost of an index theorem due to Harte \cite{HARTE}, \cite{HARTE2} and technique of Banach space embeddings introduced by the first author \cite{OPERATORTHEORY}.