Fredholmness and Weylness of block operator matrices

Abstract

This paper has aim to characterize Fredholmness and Weylness of upper triangular operator matrices having arbitrary dimension $n\geq 2$. We present various characterization results in the setting of infinite dimensional Hilbert spaces, thus extending some known results from Cao X. et al. (Acta Math. Sin. (Engl. Ser.) \textbf{22} (2006), no. 1, 169–178 and J. Math. Anal. Appl. \textbf{304} (2005), no. 2, 759–771) and Zhang et al. (J. Math. Anal. Appl. \textbf{392} (2012), no. 2, 103–110) to the case of arbitrary dimension $n\geq2$. We pose our results without using separability assumption, thus improving perturbation results from Wu X. et al. (Ann. Funct. Anal. \textbf{11} (2020), no. 3, 780–798 and Acta Math. Sin. (Engl. Ser.) 36 (2020), no. 7, 783–796).