• Journal of applied mathematics & informatics
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• 제5권2호
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• pp.433-448
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• 1998

In this study we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a par-tially ordered Banach space. In this way the metric properties of the examined problem can be analyzed more precisely. Moreover this approach allows us to derive from the same theorem on the one hand semi-local results of kantorovich-type and on the other hand 2global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved on the other hand by choosing our operators appropriately we can find sharper error bounds on the distances involved than before. Furthermore we show that special cases of our results reduce to the corresponding ones already in the literature. Finally our results are used to solve integral equations that cannot be solved with existing methods.

• Kyungpook Mathematical Journal
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• 제48권2호
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• pp.281-286
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• 2008

In this note we prove that if A and $B^*$ are subnormal operators and is a bounded linear operator such that AX - XB is a Hilbert-Schmidt operator, then f(A)X - Xf(B) is also a Hilbert-Schmidt operator and $${\parallel}f(A)X\;-\;Xf(B){\parallel}_2\;\leq\;L{\parallel}AX\;-\;XB{\parallel}_2$$, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and $X\;{\in}\;\cal{L}(\cal{H})$ is such that SX - XT belongs to a norm ideal (J, ${\parallel}\;{\cdot}\;{\parallel}_J$) and prove that f(S)X - Xf(T) $\in$ J and ${\parallel}f(S)X\;-\;Xf(T){\parallel}_J\;\leq\;C{\parallel}SX\;-\;XT{\parallel}_J$, for f in a certain class of functions.

• 대한수학회보
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• 제50권1호
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• pp.161-173
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• 2013

This paper is concerned with the space $\mathcal{K}_{w^*}(X^*,Y)$ of $weak^*$ to weak continuous compact operators from the dual space $X^*$ of a Banach space X to a Banach space Y. We show that if $X^*$ or $Y^*$ has the Radon-Nikod$\acute{y}$m property, $\mathcal{C}$ is a convex subset of $\mathcal{K}_{w^*}(X^*,Y)$ with $0{\in}\mathcal{C}$ and T is a bounded linear operator from $X^*$ into Y, then $T{\in}\bar{\mathcal{C}}^{{\tau}_{\mathcal{c}}}$ if and only if $T{\in}\bar{\{S{\in}\mathcal{C}:{\parallel}S{\parallel}{\leq}{\parallel}T{\parallel}\}}^{{\tau}_{\mathcal{c}}}$, where ${\tau}_{\mathcal{c}}$ is the topology of uniform convergence on each compact subset of X, moreover, if $T{\in}\mathcal{K}_{w^*}(X^*, Y)$, here $\mathcal{C}$ need not to contain 0, then $T{\in}\bar{\mathcal{C}}^{{\tau}_{\mathcal{c}}}$ if and only if $T{\in}\bar{\mathcal{C}}$ in the topology of the operator norm. Some properties of $\mathcal{K}_{w^*}(X^*,Y)$ are presented.

• 대한수학회지
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• 제43권2호
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• pp.399-411
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• 2006

Let L be a subspace lattice on a Hilbert space H and X and Y be operators acting on a Hilbert space H. Let P be the projection onto $\frac\;{R(X)}$, where RX is the range of X. If PE = EP for each $E\;\in\;L$, then there exists an operator A in AlgL such that AX = Y if and only if $$sup\{{\parallel}E^{\bot}Yf{\parallel}/{\parallel}E^{\bot}Xf{\parallel}\;:\;f{\in}H,\; E{\in}L}=K\;<\;\infty$$ Moreover, if the necessary condition holds, then we may choose an operator A such that AX = Y and ${\parallel}A{\parallel} = K.$ Let x and y be vectors in H and let $P_x$ be the projection onto the singlely generated space by x. If $P_xE = EP_x$ for each $E\inL$, then the assertion that there exists an operator A in AlgL such that Ax = y is equivalent to the condition $$K_0\;:\;=\;sup\{{\parallel}E^{\bot}y{\parallel}/{\parallel}E^{\bot}x\;:\;E{\in}L}=<\;\infty$$ Moreover, we may choose an operator A such that ${\parallel}A{\parallel} = K_0$ whose norm is $K_0$ under this case.

• 대한수학회논문집
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• 제24권3호
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• pp.381-393
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• 2009

Strong convergence theorems on viscosity approximation methods for finding a common zero of a finite family accretive operators are established in a reflexive and strictly Banach space having a uniformly G$\hat{a}$teaux differentiable norm. The main theorems supplement the recent corresponding results of Wong et al. [29] and Zegeye and Shahzad [32] to the viscosity method together with different control conditions. Our results also improve the corresponding results of [9, 16, 18, 19, 25] for finite nonexpansive mappings to the case of finite pseudocontractive mappings.

• 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.

• 대한수학회논문집
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• 제26권3호
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• pp.483-498
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• 2011

In this paper, we introduce a new class of uniformly point-wise R-subweakly commuting self-mappings and prove several common fixed point theorems and best approximation results for uniformly point-wise R-subweakly commuting asymptotically I-nonexpansive mappings in normed linear spaces. We also establish some results concerning strong convergence of nearest common fixed points of asymptotically I-non-expansive mappings in reflexive Banach spaces with a uniformly G$\^{a}$teaux differentiable norm. Our results unify and generalize various known results given by some authors to a more general class of noncommuting mappings.

• 대한수학회보
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• 제56권5호
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• pp.1273-1283
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• 2019

In matrix analysis, the Wielandt-Mirsky conjecture states that $$dist({\sigma}(A),{\sigma}(B)){\leq}{\parallel}A-B{\parallel}$$ for any normal matrices $A,B{\in}{\mathbb{C}}^{n{\times}n}$ and any operator norm ${\parallel}{\cdot}{\parallel}$ on $C^{n{\times}n}$. Here dist(${\sigma}(A),{\sigma}(B)$) denotes the optimal matching distance between the spectra of the matrices A and B. It was proved by A. J. Holbrook (1992) that this conjecture is false in general. However it is true for the Frobenius distance and the Frobenius norm (the Hoffman-Wielandt inequality). The main aim of this paper is to study the Hoffman-Wielandt inequality and some weaker versions of the Wielandt-Mirsky conjecture for matrix polynomials.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제5권2호
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• pp.109-114
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• 1998

Let H be a separable complex H be a space and let (equation omitted)(H) be the *-algebra of all bounded linear operators on H. An operator T in (equation omitted)(H) is said to be p-hyponormal if ($T^{\ast}T)^p - (TT^{\ast})^{p}\geq$ 0 for 0 ＜ p ＜ 1. If p = 1, T is hyponormal and if p = $\frac{1}{2}$, T is semi-hyponormal. In this paper, by using a technique introduced by S. K. Berberian, we show that the approximate point spectrum $\sigma_{\alpha p}(T) of a pure p-hyponormal operator T is empty, and obtains the compact perturbation of T.

• Journal of the Korean Statistical Society
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• 제21권2호
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• pp.153-166
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• 1992

Consider an unknown regression function f of the response Y on a d-dimensional measurement variable X. It is assumed that f belongs to a tensor Sobolev space. Let T denote a differential operator. Let $\hat{T}_n$ denote an estimator of T(f) based on a random sample of size n from the distribution of (X, Y), and let $\Vert \hat{T}_n - T(f) \Vert_2$ be the usual $L_2$ norm of the restriction of $\hat{T}_n - T(f)$ to a subset of $R^d$. Under appropriate regularity conditions, the optimal rate of convergence for $\Vert \hat{T}_n - T(f) \Vert_2$ is discussed.