• East Asian mathematical journal
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• 제31권5호
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• pp.695-706
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• 2015

Let ${\mathcal{H}}$ be an infinite dimensional complex Hilbert space, and let $B({\mathcal{H}})$ be the algebra of all bounded linear operators on ${\mathcal{H}}$. Recall that an operator $T{\in}B({\mathcal{H})$ has property B(n) if ${\mid}T^n{\mid}{\geq}{\mid}T{\mid}^n$, $n{\geq}2$, which generalizes the class A-operator. We characterize the property B(n) of weighted shifts $S_{\lambda}$ over (${\eta},\;{\kappa}$)-type directed trees which appeared in the study of subnormality of weighted shifts over directed trees recently. In addition, we discuss the property B(n) of weighted shifts $S_{\lambda}$ over (2, 1)-type directed trees with nonzero weights are being distinct with respect to $n{\geq}2$. And we give some properties of weighted shifts $S_{\lambda}$ over (2, 1)-type directed trees with property B(2).

• 대한수학회논문집
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• 제27권1호
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• pp.77-95
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• 2012

For a bounded linear operator T we prove the following assertions: (a) If T is algebraically (p, k)-quasihyponormal, then T is a-isoloid, polaroid, reguloid and a-polaroid. (b) If $T^*$ is algebraically (p, k)-quasihyponormal, then a-Weyl's theorem holds for f(T) for every $f{\in}Hol({\sigma}T))$, where $Hol({\sigma}(T))$ is the space of all functions that analytic in an open neighborhoods of ${\sigma}(T)$ of T. (c) If $T^*$ is algebraically (p, k)-quasihyponormal, then generalized a-Weyl's theorem holds for f(T) for every $f{\in}Hol({\sigma}T))$. (d) If T is a (p, k)-quasihyponormal operator, then the spectral mapping theorem holds for semi-B-essential approximate point spectrum $\sigma_{SBF_+^-}(T)$, and for left Drazin spectrum ${\sigma}_{lD}(T)$ for every $f{\in}Hol({\sigma}T))$.

• 대한수학회보
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• 제31권2호
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• pp.173-180
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• 1994

Let H be a separable, infinite dimensional, compled Hilbert space and let L(H) be the algebra of all bounded linear operators on H. A dual algebra is a subalgebra of L(H) that contains the identity operator $I_{H}$ and is closed in the ultraweak topology on L(H). Note that the ultraweak operator topology coincides with the wea $k^{*}$ topology on L(H)(see [3]). Bercovici-Foias-Pearcy [3] studied the problem of solving systems of simultaneous equations in the predual of a dual algebra. The theory of dual algebras has been applied to the topics of invariant subspaces, dilation theory and reflexibity (see [1],[2],[3],[5],[6]), and is deeply related with properties ( $A_{m,n}$). Jung-Lee-Lee [7] introduced n-separating sets for subalgebras and proved the relationship between n-separating sets and properties ( $A_{m,n}$). In this paper we will study the relationship between direct sum and properties ( $A_{m,n}$). In particular, using some results of [7] we obtain relationship between n-separating sets and direct sum of von Neumann algebras.ras.s.ras.

• 대한수학회보
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• 제41권3호
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• pp.457-464
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• 2004

Let X be a Banach space and Z a closed subspace of a Banach space Y. Denote by Ｌ(X, Y) the space of all bounded linear operators from X to Y and by Ｋ(X, Y) its subspace of compact linear operators. Using Hahn-Banach extension operators corresponding to ideal projections, we prove that if either $X^{**}$ or $Y^{*}$ has the Radon-Nikodym property and Ｋ(X, Y) is an M-ideal (resp. an HB-subspace) in Ｌ(X, Y), then Ｋ(X, Z) is also an M-ideal (resp. HB-subspace) in Ｌ(X, Z). If Ｌ(X, Y) has property SU instead of being an M-ideal in Ｌ(X, Y) in the above, then Ｋ(X, Z) also has property SU in Ｌ(X, Z). If X is a Banach space such that $X^{*}$ has the metric compact approximation property with adjoint operators, then M-ideal (resp. HB-subspace) property of Ｋ(X, Y) in Ｌ(X, Y) is inherited to Ｋ(X, Z) in Ｌ(X, Z).

• 대한수학회지
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• 제57권1호
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• pp.171-194
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• 2020

For a nonautonomous dynamics defined by a sequence of bounded linear operators on a Banach space, we give a characterization of the existence of an exponential dichotomy with respect to a sequence of norms in terms of the invertibility of a certain linear operator between general admissible spaces. This notion of an exponential dichotomy contains as very special cases the notions of uniform, nonuniform and tempered exponential dichotomies. As applications, we detail the consequences of our results for the class of tempered exponential dichotomies, which are ubiquitous in the context of ergodic theory, and we show that the notion of an exponential dichotomy under sufficiently small parameterized perturbations persists and that their stable and unstable spaces are as regular as the perturbation.

• International Journal of Control, Automation, and Systems
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• 제6권1호
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• pp.24-34
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• 2008

A novel neural network model, termed the standard neural network model (SNNM), similar to the nominal model in linear robust control theory, is suggested to facilitate the synthesis of controllers for delayed (or non-delayed) nonlinear systems composed of neural networks. The model is composed of a linear dynamic system and a bounded static delayed (or non-delayed) nonlinear operator. Based on the global asymptotic stability analysis of SNNMs, Static state-feedback controller and dynamic output feedback controller are designed for the SNNMs to stabilize the closed-loop systems, respectively. The control design equations are shown to be a set of linear matrix inequalities (LMIs) which can be easily solved by various convex optimization algorithms to determine the control signals. Most neural-network-based nonlinear systems with time delays or without time delays can be transformed into the SNNMs for controller synthesis in a unified way. Two application examples are given where the SNNMs are employed to synthesize the feedback stabilizing controllers for an SISO nonlinear system modeled by the neural network, and for a chaotic neural network, respectively. Through these examples, it is demonstrated that the SNNM not only makes controller synthesis of neural-network-based systems much easier, but also provides a new approach to the synthesis of the controllers for the other type of nonlinear systems.

• 대한수학회논문집
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• 제32권3호
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• pp.661-676
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• 2017

Let T be a bounded linear operator acting on a complex Hilbert space ${\mathfrak{H}}$. In this paper we introduce the class, denoted ${\mathcal{Q}}(A(k),m)$, of operators satisfying $T^{m{\ast}}(T^{\ast}{\mid}T{\mid}^{2k}T)^{1/(k+1)}T^m{\geq}T^{{\ast}m}{\mid}T{\mid}^2T^m$, where m is a positive integer and k is a positive real number and we prove basic structural properties of these operators. Using these results, we prove that if P is the Riesz idempotent for isolated point ${\lambda}$ of the spectrum of $T{\in}{\mathcal{Q}}(A(k),m)$, then P is self-adjoint, and we give a necessary and sufficient condition for $T{\otimes}S$ to be in ${\mathcal{Q}}(A(k),m)$ when T and S are both non-zero operators. Moreover, we characterize the quasinilpotent part $H_0(T-{\lambda})$ of class A(k) operator.

• 대한수학회보
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• 제38권4호
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• pp.663-668
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• 2001

X and Y are Banach spaces for which K(X, Y), the space of compact operators from X to Y, is an M-ideal in L(X, Y), the space of bounded linear operators form X to Y. If Z is a closed subspace of Y such that L(X, Z) has property SU in L(X, Y) and d(T, K(X, Z)) = d(T, K(X, Y)) for all $T \in L(X, Z)$, then K(X, Z) is an M-ideal in L(X, Z) if and only if it has property SU is L(X, Z).

• 호남수학학술지
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• 제28권4호
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• pp.533-541
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• 2006

let E be a Finsler modules over $C^*$-algebras. A with norm-map $\rho$ and L(E) set of all A-linear bonded operators on E. We show that the canonical homomorphism ${\phi}:L(E){\rightarrow}L(E_I)$ sending each operator T to its restriction $T|E_I$ is injective if and only if I is an essential ideal in the underlying $C^*$-algebra A. We also show that $T{\in}L(E)$ is a bounded below if and only if ${\mid}{\mid}x{\mid}{\mid}={\mid}{\mid}{\rho}{\prime}(x){\mid}{\mid}$ is complete, where ${\rho}{\prime}(x)={\rho}(Tx)$ for all $x{\in}E$. Also, we give a necessary and sufficient condition for the equivalence of the norms generated by the norm map.

• 대한수학회지
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• 제52권1호
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• pp.177-190
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• 2015

Let X be a completely regular Hausdorff space, and E and F be Banach spaces. Let $C_{rc}(X,E)$ be the Banach space of all continuous functions $f:X{\rightarrow}E$ such that f(X) is a relatively compact set in E. We establish an integral representation theorem for bounded linear operators $T:C_{rc}(X,E){\rightarrow}F$. We characterize continuous operators from $C_{rc}(X,E)$, provided with the strict topologies ${\beta}_z(X,E)$ ($z={\sigma},{\tau}$) to F, in terms of their representing operator-valued measures.