• Bulletin of the Korean Mathematical Society
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• v.46 no.1
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• pp.117-123
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• 2009

This paper is a study of some properties of a collection of bounded linear operators resulting from terraced matrices M acting through multiplication on ${\ell}^2$; the term terraced matrix refers to a lower triangular infinite matrix with constant row segments. Sufficient conditions are found for M to be posinormal, meaning that $MM^*=M^*PM$ for some positive operator P on ${\ell}^2$; these conditions lead to new sufficient conditions for the hyponormality of M. Sufficient conditions are also found for the adjoint $M^*$ to be posinormal, and it is observed that, unless M is essentially trivial, $M^*$ cannot be hyponormal. A few examples are considered that exhibit special behavior.

• Kyungpook Mathematical Journal
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• v.46 no.1
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• pp.89-96
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• 2006

For a rank-1 matrix A, there is a factorization as $A=ab^t$, the product of two vectors a and b. We characterize the linear operators that preserve rank and some equivalent condition of rank-1 matrices over a chain semiring. We also obtain a linear operator T preserves the rank of rank-1 matrices if and only if it is a form (P, Q, B)-operator with appropriate permutation matrices P and Q, and a matrix B with all nonzero entries.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.821-835
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• 2023

Hankel operators and their variants have abundant applications in numerous fields. For a non-zero complex number r, the r-Hankel operators on a Hilbert space 𝓗 define a class of one such variant. This article introduces and explores some properties of two other variants of Hankel operators namely k^th-order (C, r)-Hankel operators and k^th-order (R, r)-Hankel operators (k ≥ 2) which are closely related to r-Hankel operators in such a way that a k^th-order (C, r)-Hankel matrix is formed from r^k-Hankel matrix on deleting every consecutive (k - 1) columns after the first column and a k^th-order (R, r^k)-Hankel matrix is formed from r-Hankel matrix if after the first column, every consecutive (k - 1) columns are deleted. For |r| ≠ 1, the characterizations for the boundedness of these operators are also completely investigated. Finally, an appropriate approach is also presented to extend these matrices to two-way infinite matrices.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1351-1370
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• 2018

In the first part of this paper we show that, under some conditions, a polynomially demicompact operator can be demicompact. An example involving the Caputo fractional derivative of order ${\alpha}$ is provided. Furthermore, we give a refinement of the left and the right Weyl essential spectra of a closed linear operator involving the class of demicompact ones. In the second part of this work we provide some sufficient conditions on the inputs of a closable block operator matrix, with domain consisting of vectors which satisfy certain conditions, to ensure the demicompactness of its closure. Moreover, we apply the obtained results to determine the essential spectra of this operator.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.973-986
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• 2020

In this paper, we compute the Hankel matrix representation of the Hankel operator on the Hardy space of a general bounded domain with respect to special orthonormal bases for the Hardy space and its orthogonal complement. Moreover we obtain the compact form of the Hankel matrix for the unit disc case with respect to these bases. One can see that the Hankel matrix generated by this computation turns out to be a generalization of the case of the unit disc from the single simply connected domain to multiply connected domains with much diversities of bases.

• The Pure and Applied Mathematics
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• v.21 no.2
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• pp.121-128
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• 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 Mathematical Society
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• v.56 no.2
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• pp.507-521
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• 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.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1139-1150
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• 2009

We obtain necessary and sufficient conditions for $2{\tims}2$ block operator valued matrices to be Moore-Penrose (MP) invertible and give new representations of such MP inverses in terms of the individual blocks.

• The Pure and Applied Mathematics
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• v.28 no.1
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• pp.43-53
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• 2021

We study translation surfaces in the Euclidean 3-space ��3 and the Gauss map N with respect to the so-called Cheng-Yau operator ☐. As a result, we prove that the only translation surfaces with Gauss map N satisfying ☐N = AN for some 3 × 3 matrix A are the flat ones. We also show that the only translation surfaces with Gauss map N satisfying ☐N = AN for some nonzero 3 × 3 matrix A are the cylindrical surfaces.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.187-200
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• 2019

Truncated Hankel operators are compressions of classical Hankel operators to model spaces. In this paper we describe matrix representations of truncated Hankel operators on finite-dimensional model spaces. We then show that the obtained descriptions hold also for some infinite-dimensional cases.