Closedness of ranges of upper-triangular operators
type: Item
instance of: scholarly article
title: Closedness of ranges of upper-triangular operators
zbMATH Open document ID: 1206.47001
author: Yanni Dou
author: Gui-Chun Du
author: Hongke Du
author: Chunfang Shao
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-10T00:00:00Z
review text: The authors consider block upper-triangular operators of the form \(M_C = \left( \begin{smallmatrix} A & C \\ 0 & B \\ \end{smallmatrix} \right) \) on separable Hilbert spaces. Results concerning the closure of the range of \(M_C\) are obtained in terms of properties of the operators \(A\) and \(B\). In particular, it is shown that if the ranges of both \(A\) and \(B\) are closed, then the closure of the range of \(M_C\) depends solely on whether or not either the dimension of the nullspace of \(A^*\) or the nullspace of \(B\) is finite. An operator \(T\) is Kato non-singular if the range of \(T\) is closed and the nullspace of \(T\) is contained in the range of \(T^n\) for all \(n \in \mathbb N\). The authors also prove that if \(A\) and \(B\) are Kato non-singular operators, then \(M_C\) is Kato non-singular if and only if either \(A\) is surjective or \(B\) is injective.
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 5563843
zbMATH Keywords: closed range
zbMATH Keywords: Kato non-singularity
zbMATH Keywords: operator matrix
zbMATH Keywords: upper-triangular operator
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.02.014
OpenAlex ID: W2077046307
cites work: Riesz points of upper triangular operator matrices
cites work: On the perturbations of spectra of upper triangular operator matrices
cites work: A note on the spectrum of an upper triangular operator matrix
cites work: Essential approximate point spectra and Weyl's theorem for operator matrices
cites work: Browder essential approximate point spectra and hypercyclicity for operator matrices
cites work: Semi-Fredholm spectrum and Weyl's theorem for operator matrices
cites work: Q5456130
cites work: Q4422033
cites work: A note on Weyl’s theorem for operator matrices
cites work: On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space
cites work: Perturbation of Spectrums of 2 × 2 Operator Matrices
cites work: Invertible completions of $2\times 2$ upper triangular operator matrices
cites work: Norm inequalities of positive operator matrices
cites work: The boundedness below of \(2\times 2\) upper triangular operator matrices
cites work: Weyl's theorem for operator matrices
cites work: Weyl spectra of operator matrices
cites work: Q5430194
cites work: Invertible completions of operator matrices
cites work: Browder spectra of upper-triangular operator matrices
DOI: 10.1016/J.JMAA.2009.02.014
description: scientific article; zbMATH DE number 5563843