• Bulletin of the Korean Mathematical Society
• /
• v.39 no.3
• /
• pp.423-430
• /
• 2002

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $AX_{}i$ = $Y_{i}$ for i/ = 1,2,…, n. In this article, we obtained the following : Let X ＝ ($x_{i\sigma(i)}$ and Y ＝ ($y_{ij}$ be operators in B(H) such that $X_{i\sigma(i)}\neq\;0$ for all i. Then the following statements are equivalent. (1) There exists an operator A in Alg L such that AX = Y, every E in L reduces A and A is a self-adjoint operator. (2) sup ${\frac{\parallel{\sum^n}_{i=1}E_iYf_i\parallel}{\parallel{\sum^n}_{i=1}E_iXf_i\parallel}n\;\epsilon\;N,E_i\;\epsilon\;L and f_i\;\epsilon\;H}$ < $\infty$ and $x_{i,\sigma(i)}y_{i,\sigma(i)}$ is real for all i = 1,2, ....

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.5
• /
• pp.991-1002
• /
• 2011

An iterative algorithm is provided to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of some variational inequality in a Hilbert space. Using this result, we consider a strong convergence result for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Our results include the previous results as special cases and can be viewed as an improvement and refinement of the previously known results.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.6
• /
• pp.1275-1283
• /
• 2010

If the Wiener-Hopf $C^*$-algebra W(G,M) for a discrete group G with a semigroup M has the uniqueness property, then the structure of it is to some extent independent of the choice of isometries on a Hilbert space. In this paper we show that if the Wiener-Hopf $C^*$-algebra W(G,M) of a partially ordered group G with the positive cone M has the uniqueness property, then (G,M) is weakly unperforated. We also prove that the Wiener-Hopf $C^*$-algebra W($\mathbb{Z}$, M) of subsemigroup generating the integer group $\mathbb{Z}$ is isomorphic to the Toeplitz algebra, but W($\mathbb{Z}$, M) does not have the uniqueness property except the case M = $\mathbb{N}$.

• Journal of the Korean Mathematical Society
• /
• v.47 no.2
• /
• pp.351-361
• /
• 2010

For a bounded linear operator T on a separable complex infinite dimensional Hilbert space $\mathcal{H}$, we say that T is a quasi-class (A, k) operator if $T^{*k}|T^2|T^k\;{\geq}\;T^{*k}|T|^2T^k$. In this paper we prove that if T is a quasi-class (A, k) operator and f is an analytic function on an open neighborhood of the spectrum of T, then f(T) satisfies Weyl's theorem. Also, we consider the tensor product for quasi-class (A, k) operators.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.2
• /
• pp.357-371
• /
• 2014

We shall show that the Riesz idempotent $E_{\lambda}$ of every *-paranormal operator T on a complex Hilbert space H with respect to each isolated point ${\lambda}$ of its spectrum ${\sigma}(T)$ is self-adjoint and satisfies $E_{\lambda}\mathcal{H}=ker(T-{\lambda})= ker(T-{\lambda})^*$. Moreover, Weyl's theorem holds for *-paranormal operators and more general for operators T satisfying the norm condition $||Tx||^n{\leq}||T^nx||\,||x||^{n-1}$ for all $x{\in}\mathcal{H}$. Finally, for this more general class of operators we find a sufficient condition such that $E_{\lambda}\mathcal{H}=ker(T-{\lambda})= ker(T-{\lambda})^*$ holds.

• The Journal of the Acoustical Society of Korea
• /
• v.22 no.3E
• /
• pp.133-140
• /
• 2003

A time series simulation is presented by a ray approach for the simulating the received waveform of a broadband acoustical signals interacting with the ocean boundaries. The environment is assumed to be horizontally stratified, and the seafloor is described in terms of homogeneous fluid half-space. The ray approach includes the effects of reflection from the air-water, water-sediment interface and phase shifts due to boundaries interaction. To generate time series, we assume that the acoustic energy propagates from source to receiver along eigenrays and represent the action of the bottom on the incident wave by a linear filter and characterized in the frequency domain by the transfer function. As example application, the time series for an explosive source in a shallow water environment is calculated and analyzed in terms of acoustical process. good agreement with measured time series is demonstrated.

• Journal of the Korean Mathematical Society
• /
• v.31 no.4
• /
• pp.659-667
• /
• 1994

Let $H$ be a separable, infinite dimensional, complex 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 operator topology on $L(H)$. Note that the ultraweak operator topology coincides with the weak topology on $L(H) (cf. [6]). Several functional analysists have studied the problem of solving systems of simultaneous equations in the predual of a dual algebra (cf. [3]). This theory is applied to the study of invariant subspaces and dilation theory, which are deeply related to the classes $A_{m,n}$ (that will be defined below) (cf. [3]). An abstract geometric criterion for dual algebras with property $(A_{\aleph_0}, {\aleph_0})$ was first given in [1].

• Journal of the Korean Mathematical Society
• /
• v.41 no.4
• /
• pp.617-627
• /
• 2004

Let (equation omitted) denote the class of bounded linear Hilbert space operators with the property that $\midA^2\mid\geq\midA\mid^2$. In this paper we show that (equation omitted)-operators are finitely ascensive and that, for non-zero operators A and B, A (equation omitted) B is in (equation omitted) if and only if A and B are in (equation omitted). Also, it is shown that if A is an operator such that p(A) is in (equation omitted) for a non-trivial polynomial p, then Weyl's theorem holds for f(A), where f is a function analytic on an open neighborhood of the spectrum of A.

• Journal of the Korean Mathematical Society
• /
• v.43 no.2
• /
• pp.357-371
• /
• 2006

Let $X\;=\;(X_t,\;t{\in}[0, T])$ be a generalized Brownian motion(gBm) determined by mean function a(t) and variance function b(t). Let $L^2({\mu})$ denote the Hilbert space of square integrable functionals of $X\;=\;(X_t - a(t),\; t {in} [0, T])$. In this paper we consider a class of nonlinear functionals of X of the form F(. + a) with $F{in}L^2({\mu})$ and discuss their analysis. Firstly, it is shown that such functionals do not enjoy, in general, the square integrability and Malliavin differentiability. Secondly, we establish regularity conditions on F for which F(.+ a) is in $L^2({\mu})$ and has its Malliavin derivative. Finally we apply these results to compute the price and the hedging portfolio of a contingent claim in our financial market model based on a gBm X.

• Journal of the Korean Mathematical Society
• /
• v.45 no.3
• /
• pp.771-780
• /
• 2008

Let $M_C=\(\array{A&C\\0&B}\)$ be a $2{\times}2$ upper triangular operator matrix acting on the Hilbert space $H{\bigoplus}K\;and\;let\;{\sigma}_w(\cdot)$ denote the Weyl spectrum. We give the necessary and sufficient conditions for operators A and B which ${\sigma}_w\(\array{A&C\\0&B}\)={\sigma}_w\(\array{A&C\\0&B}\)\;or\;{\sigma}_w\(\array{A&C\\0&B}\)={\sigma}_w(A){\cup}{\sigma}_w(B)$ holds for every $C{\in}B(K,\;H)$. We also study the Weyl's theorem for operator matrices.