Closedness of ranges of upper-triangular operators

Abstract

In this paper, the closedness of ranges and Kato non-singularity of upper triangular operator matrices are investigated.Crown

Completions of upper-triangular matrices to Kato nonsingularity

“…Throughout the paper we will make extensive use of a variant of a particular result from [7] on the completion problem to closed range operator matrices. One readily checks that both the statement and the proof of Theorem 2.5 from [7] remain valid (the proof is identical to that given in [7]) if A ∈ B(H 1 , H 2 ) and B ∈ B(K 1 , K 2 ), for Hilbert spaces H 1 , H 2 , K 1 and K 2 , where neither H 1 and H 2 nor K 1 and K 2 are assumed to be the same. More precisely, we have the following result.…”

“…Some particular cases of the completion problem to Kato nonsingularity in the case A is Kato nonsingular were already considered in [2] and [7]. The following two theorems, which we will often use in the sequel, were proved: In the remark below and throughout the rest of the paper, for A ∈ B(H) and B ∈ B(K), we will use the following notation:…”

“…In [2] M. Barraa and M. Boumazgour considered the completion problem to Kato nonsingular M C and gave some sufficient conditions for the existence of such operators C. These conditions always assumed, among other things, Kato nonsigularity of A or injectivity of B. Later Dou et al [7] generalized some of their results. In this paper, we completely solve the problem of completion to Kato nonsingularity in each of the following cases:…”

Extension of the GSMW Formula in Weaker Assumptions

“…Because of its wide applications in statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations (see [5]), the properties and generalizations of the SMW formula have caught mathematicians attention (see [1][2][3][4][5][6][7][8]). Duan (see [9]) finally generalized the SMW formula to the {2}-inverse (hence, to all the inverses, uniformly denoted by ⊙ ).…”

A Generalization of the SMW Formula of OperatorA+YGZ*to the{2}-Inverse Case

“…There are numerous applications of the SMW formula in various fields (see [1][2][3][4][5][6]12]). An excellent review by Hager [3] described some of the applications to statistics networks, structural analysis, asymptotic analysis, optimization, and partial differential equations.…”