  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, $n>3$.
