• 대한수학회지
• /
• 제57권4호
• /
• pp.893-913
• /
• 2020

Let 𝓢 be the collection of the operator matrices $\(\array{A&C\\Z&B}\)$ where the range of C is closed. In this paper, we study the properties of operator matrices in the class 𝓢. We first explore various local spectral relations, that is, the property (β), decomposable, and the property (C) between the operator matrices in the class 𝓢 and their component operators. Moreover, we investigate Weyl and Browder type spectra of operator matrices in the class 𝓢, and as some applications, we provide the conditions for such operator matrices to satisfy a-Weyl's theorem and a-Browder's theorem, respectively.

• Kyungpook Mathematical Journal
• /
• 제59권2호
• /
• pp.225-231
• /
• 2019

In a previous paper, the authors of this paper studied $2{\times}2$ matrices in upper triangular form, whose entries are operators on Hilbert spaces, and in which the the (1, 1) entry has a nontrivial hyperinvariant subspace. We were able to show, in certain cases, that the $2{\times}2$ matrix itself has a nontrivial hyperinvariant subspace. This generalized two earlier nice theorems of H. J. Kim from 2011 and 2012, and made some progress toward a solution of a problem that has been open for 45 years. In this paper we continue our investigation of such $2{\times}2$ operator matrices, and we improve our earlier results, perhaps bringing us closer to the resolution of the long-standing open problem, as mentioned above.

• 대한수학회보
• /
• 제37권4호
• /
• pp.691-696
• /
• 2000

Equations. See Full-text

• 대한수학회지
• /
• 제56권2호
• /
• pp.507-521
• /
• 2019

For any $m{\times}n$ nonbinary Boolean matrix A, its spanning column rank is the minimum number of the columns of A that spans its column space. We have a characterization of spanning column rank-1 nonbinary Boolean matrices. We investigate the linear operators that preserve the spanning column ranks of matrices over the nonbinary Boolean algebra. That is, for a linear operator T on $m{\times}n$ nonbinary Boolean matrices, it preserves all spanning column ranks if and only if there exist an invertible nonbinary Boolean matrix P of order m and a permutation matrix Q of order n such that T(A) = PAQ for all $m{\times}n$ nonbinary Boolean matrix A. We also obtain other characterizations of the (spanning) column rank preserver.

• 대한수학회보
• /
• 제53권3호
• /
• pp.681-698
• /
• 2016

In this paper, we investigate relative essential spectra of $2{\times}2$ block operator matrix using the Fredholm perturbation theory. Furthermore, an example for two-group transport equations is presented to illustrate the validity of the main results.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제21권2호
• /
• pp.121-128
• /
• 2014

The Boolean rank of a nonzero m $m{\times}n$ Boolean matrix A is the least integer k such that there are an $m{\times}k$ Boolean matrix B and a $k{\times}n$ Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank k with $1{\leq}k{\leq}min\{m,n\}$.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제2권2호
• /
• pp.27-30
• /
• 1998

Let A and B be $m{\times}n$ matrices. A linear operator T preserves the set of matrices on which the rank is additive if rank(A+B) = rank(A)+rank(B) implies that rank(T(A) + T(B)) = rankT(A) + rankT(B). We characterize the set of all linear operators which preserve the set of pairs of $n{\times}n$ matrices on which the rank is additive.

• 대한수학회보
• /
• 제46권2호
• /
• pp.373-385
• /
• 2009

An m${\times}$n Boolean matrix A is called regular if there exists an n${\times}$m Boolean matrix X such that AXA = A. We have characterizations of Boolean regular matrices. We also determine the linear operators that strongly preserve Boolean regular matrices.

• 대한수학회보
• /
• 제45권2호
• /
• pp.355-363
• /
• 2008

For a Boolean rank 1 matrix $A=ab^t$, we define the perimeter of A as the number of nonzero entries in both a and b. The perimeter of an $m{\times}n$ Boolean matrix A is the minimum of the perimeters of the rank-1 decompositions of A. In this article we characterize the linear operators that preserve the perimeters of Boolean matrices.

• 충청수학회지
• /
• 제4권1호
• /
• pp.11-38
• /
• 1991

Let $E_n$ be the real ordered space of all $n{\times}n$ Hermitian Matrices and let T be a positive linear operator from $E_2$ to $E_3$. We prove in this paper that T is extreme if and only if T is unitarily equivalent to a map of the form $S_z$ for some $z{\in}C^2$ where $S_z$ is defined by $S_z(xx^*)=ww^*$, $w_i=x_iz_i$ for i = 1, 2 and $w_3=0$.