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).