This paper has aim to characterize Fredholmness and Weylness of upper triangular operator matrices having arbitrary dimension n ? 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.) 22 (2006), no. 1, 169-178 and J. Math. Anal. Appl. 304 (2005), no. 2, 759-771) and Zhang et al. (J. Math. Anal. Appl. 392 (2012), no. 2, 103-110) to the case of arbitrary dimension n ? 2. We pose our results without using separability assumption, thus improving perturbation results fromWu X. et al. (Ann. Funct. Anal. 11 (2020), no. 3, 780-798 and Acta Math. Sin. (Engl. Ser.) 36 (2020), no. 7, 783-796).

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Let T d n (A) be an upper triangular operator matrix of dimension n with given diagonal entries. We assume T d n (A) acts on a direct topological sum of Banach spaces, where tuple A = (A ij ) 1≤i

This paper deals with characterizations of Fredholmness and Weylness of upper triangular operator matrices of an arbitrary dimension n ≥ 2. We present various characterization results in the settings of separable and arbitrary infinite dimensional Hilbert spaces, thus extending some known results from [3], [4], [24] to case of arbitrary n ≥ 2. As corollaries, we get corrected perturbation results from [21], [22], [24]. Namely, we show that certain perturbation results for upper triangular operators of dimension n > 2 can not hold with an equality, but only with a double inclusion instead, and we explain why closedness of ranges of diagonal operators must not be neglected in the results of this kind. Finally, we give a comment on possible extensions to infinite dimensional Banach spaces.

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