Completion problem of upper triangular $3\times3$ operator matrices on arbitrary Banach spaces

We solve the completion problem of $3\times3$ upper triangular operator matrix acting on a direct sum of Banach spaces and hence generalize the famous result of Han, Lee, Lee (Proc. Amer. Math. Soc. 128 (1) (2000), 119-123) to a greater dimension of a matrix. Our main tools are Harte's ghost of an index theorem and Banach spaces embeddings. We overcome the lack of orthogonality in Banach spaces by exploiting decomposition properties of inner regular operators, and of Fredholm regular operators when needed. Finally, we provide some necessity results related to the invertibility of $n\times n$ upper triangular operators, $n>3$.