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

TL;DR

This paper addresses the completion problem of 3x3 upper triangular operator matrices on Banach spaces, extending previous results to higher dimensions using advanced operator theory tools.

Contribution

It generalizes the completion problem for 3x3 upper triangular operator matrices on Banach spaces, employing Harte's ghost index theorem and space embeddings.

Findings

Solved the 3x3 matrix completion problem on Banach spaces.

Extended results to higher-dimensional upper triangular operators.

Provided necessary conditions for invertibility of larger matrices.

Abstract

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, $n>3$.