• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.1
• /
• pp.19-34
• /
• 2011

A bounded linear operator T on a complex Banach space X is said to have property (I) provided that T has Bishop's property (${\beta}$) and there exists an integer p > 0 such that for a closed subset F of ${\mathbb{C}}$ ${X_T}(F)={E_T}(F)=\bigcap_{{\lambda}{\in}{\mathbb{C}}{\backslash}F}(T-{\lambda})^PX$ for all closed sets $F{\subseteq}{\mathbb{C}}$, where $X_T$(F) denote the analytic spectral subspace and $E_T$(F) denote the algebraic spectral subspace of T. Easy examples are provided by normal operators and hyponormal operators in Hilbert spaces, and more generally, generalized scalar operators and subscalar operators in Banach spaces. In this paper, we prove that if T has property (I), then the quasi-nilpotent part $H_0$(T) of T is given by $$KerT^P=\{x{\in}X:r_T(x)=0\}={\bigcap_{{\lambda}{\neq}0}(T-{\lambda})^PX$$ for all sufficiently large integers p, where ${r_T(x)}=lim\;sup_{n{\rightarrow}{\infty}}{\parallel}T^nx{\parallel}^{\frac{1}{n}}$. We also prove that if T has property (I) and the spectrum ${\sigma}$(T) is finite, then T is algebraic. Finally, we prove that if $T{\in}L$(X) has property (I) and has decomposition property (${\delta}$) then T has a non-trivial invariant closed linear subspace.

• Honam Mathematical Journal
• /
• v.7 no.1
• /
• pp.1-6
• /
• 1985

Bounded linear operators defined from a space with vector norm into (LF)-complete vector lattice are represented by Hellinger integration.

• Journal of the Chungcheong Mathematical Society
• /
• v.3 no.1
• /
• pp.13-25
• /
• 1990

• Journal of the Korean Mathematical Society
• /
• v.44 no.1
• /
• pp.169-178
• /
• 2007

We consider the set of $n{\times}n$ idempotent matrices and we characterize the linear operators that preserve idempotent matrices over Boolean algebras. We also obtain characterizations of linear operators that preserve idempotent matrices over a chain semiring, the nonnegative integers and the nonnegative reals.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.2 no.2
• /
• 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.

• Journal of applied mathematics & informatics
• /
• v.26 no.3_4
• /
• pp.643-652
• /
• 2008

We construct the sets of Boolean matrix pairs, which are naturally occurred at the extreme cases for the Boolean rank inequalities relative to the sums and difference of two Boolean matrices or compared between their Boolean ranks and their real ranks. For these sets, we consider the linear operators that preserve them. We characterize those linear operators as T(X) = PXQ or $T(X)\;=\;PX^tQ$ with appropriate invertible Boolean matrices P and Q.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.5
• /
• pp.1579-1586
• /
• 2015

In this paper we investigate the barrellednes of some spaces of X-valued measures, X being a barrelled normed space, and provide examples of non barrelled spaces of bounded linear operators from a Banach space X into a barrelled normed space Y, equipped with the uniform convergence topology.

• Bulletin of the Korean Mathematical Society
• /
• v.45 no.2
• /
• 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.

• Journal of applied mathematics & informatics
• /
• v.26 no.5_6
• /
• pp.1263-1272
• /
• 2008

Let T and S be bounded linear operators on Banach spaces X and y, respectively. A linear map A : X ${\rightarrow}$ y is said to be an intertwiner if AT - SA = 0. In this paper we study the relation between local spectral properties of T and S on the assumption of AT - SA = 0. We give some example of intertwiner with T and S.

• Journal of the Chungcheong Mathematical Society
• /
• v.15 no.1
• /
• pp.53-60
• /
• 2002

In this paper, we prove the generalized Hyers-Ulam-Rassias stability of linear operators in Banach modules over a unital $C^*$-algebra associated with its unitary group.