Upper Triangular Operator Matrices and Stability of Their Various Spectra

Abstract

Denote by \(T_n^d(A)\) an upper triangular operator matrix of dimension \(n\in \mathbb {N}\) whose diagonal entries \(D_i,\ 1\le i\le n\), are known, and \(A=(A_{ij})_{1\le i<j\le n}\) is an unknown tuple of operators. This article is aimed at investigation of defect spectrum \(\mathcal {D}^{\sigma _*}=\bigcup _{i=1}^n\sigma _*(D_i){\setminus }\sigma _*(T_n^d(A))\), where \(\sigma _*\) is a spectrum corresponding to various types of invertibility: (left, right) invertibility, (left, right) Fredholm invertibility, left/right Weyl invertibility. We give characterizations for each of the previous types, and provide some sufficent conditions for the stability of certain spectrum (the case \(\mathcal {D}^{\sigma _*}=\emptyset \)). The results are proved for all matrix dimensions \(n\ge 2\), and they hold in arbitrary Hilbert spaces without assuming separability, thus generalizing results from Wu and Huang (Ann Funct Anal 11(3):780–798, 2020; Acta Math Sin 36(7):783–796, 2020). We also retrieve a result from Bai et al. (J Math Anal Appl 434(2):1065–1076, 2016) in the case \(n=2\), and we provide a precise form of the well known ‘filling in holes’ result from Han et al. (Proc Am Math Soc 128(1):119–123, 2000).