• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.529-541
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• 2005

An operator T belonging to the algebra B(H) of bounded linear transformations on a Hilbert H into itself is said to be posinormal if there exists a positive operator $P{\in}B(H)$ such that $TT^*\;=\;T^*PT$. A posinormal operator T is said to be conditionally totally posinormal (resp., totally posinormal), shortened to $T{\in}CTP(resp.,\;T{\in}TP)$, if to each complex number, $\lambda$ there corresponds a positive operator $P_\lambda$ such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P_{\lambda}^{\frac{1}{2}}(T-{\lambda}I)|^{2}$ (resp., if there exists a positive operator P such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P^{\frac{1}{2}}(T-{\lambda}I)|^{2}\;for\;all\;\lambda)$. This paper proves Weyl's theorem type results for TP and CTP operators. If $A\;{\in}\;TP$, if $B^*\;{\in}\;CTP$ is isoloid and if $d_{AB}\;{\in}\;B(B(H))$ denotes either of the elementary operators $\delta_{AB}(X)\;=\;AX\;-\;XB\;and\;\Delta_{AB}(X)\;=\;AXB\;-\;X$, then it is proved that $d_{AB}$ satisfies Weyl's theorem and $d^{\ast}_{AB}\;satisfies\;\alpha-Weyl's$ theorem.

• Proceedings of the Korean Statistical Society Conference
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• 2002.05a
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• pp.55-60
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• 2002

We present an extension of the Wong-Zakai type approximation theorem for a multiple stochastic integral. Using a piecewise linear approximation $W^{(n)}$ of a Wiener process W, we prove that the multiple integral processes {${\int}_{0}^{t}{\cdots}{\int}_{0}^{t}f(t_{1},{\cdots},t_{m})W^{(n)}(t_{1}){\cdots}W^{(n)}(t_{m}),t{\in}[0,T]$} where f is a given symmetric function in the space $C([0,T]^{m})$, converge to the multiple Stratonovich integral of f in the uniform $L^{2}$-sense.

• Honam Mathematical Journal
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• v.34 no.3
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• pp.311-326
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• 2012

The purpose of this paper is to introduce and investigate two new classes of generalized Bernoulli and Apostol-Bernoulli polynomials based on the definition given recently by the authors [29]. In particular, we obtain a new addition formula for the new class of the generalized Bernoulli polynomials. We also give an extension and some analogues of the Srivastava-Pint$\acute{e}$r addition theorem [28] for both classes. Finally, by making use of the new adition formula, we exhibit several interesting relationships between generalized Bernoulli polynomials and other polynomials or special functions.

• Journal of the Korean Mathematical Society
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• v.42 no.2
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• pp.203-222
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• 2005

Thompson series is a Hauptmodul for a genus zero group which lies between $\Gamma$o(N) and its normalizer in PSL2(R) ([1]). We construct explicit ring class fields over an imaginary quadratic field K from the Thompson series $T_g$($\alpha$) (Theorem 4), which would be an extension of [3], Theorem 3.7.5 (2) by using the Shimura theory and the standard results of complex multiplication. Also we construct various class fields over K, over a CM-field K (${\zeta}N + {\zeta}_N^{-1}$), and over a field K (${\zeta}N$). Furthermore, we find an explicit formula for the conjugates of Tg ($\alpha$) to calculate its minimal polynomial where $\alpha$ (${\in}{\eta}$) is the quotient of a basis of an integral ideal in K.

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

• Communications of the Korean Mathematical Society
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• v.28 no.2
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• pp.285-295
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• 2013

In this note we present sufficient conditions for the existence of Radon-Nikodym derivatives (RND) of operator valued measures with respect to scalar measures. The RND is characterized by the Bochner integral in the strong operator topology of a strongly measurable operator valued function with respect to a nonnegative finite measure. Using this result we also obtain a characterization of compact sets in the space of operator valued measures. An extension of this result is also given using the theory of Pettis integral. These results have interesting applications in the study of evolution equations on Banach spaces driven by operator valued measures as structural controls.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1001-1015
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• 2010

Let M be a $C^{\infty}$ real hypersurface in $\mathbb{C}^{n+1}$, $n\;{\geq}\;1$, locally given as the zero locus of a $C^{\infty}$ real valued function r that is defined on a neighborhood of the reference point $P\;{\in}\;M$. For each k = 1,..., n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n - k at P. The problem is to find an integral manifold of the real 1-form $i{\partial}r$ on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.

• Korean Journal of Mathematics
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• v.16 no.4
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• pp.583-587
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• 2008

Let $\mathfrak{L(H)}$ denote the algebra of bounded linear operators on a separable infinite dimensional complex Hilbert space $\mathfrak{H}$. We say that $T{\in}\mathfrak{L(H)}$ is a quasi-class A operator if $$T^*{\mid}T^2{\mid}T{{\geq}}T^*{\mid}T{\mid}^2T$$. In this paper we prove that if A and B are quasi-class A operators, and $B^*$ is invertible, then for a Hilbert-Schmidt operator X $$AX=XB\;implies\;A^*X=XB^*$$.

• The Pure and Applied Mathematics
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• v.15 no.4
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• pp.387-392
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• 2008

In this note, an alternative proof and extensions are provided for the following conclusions in [6, Theorem 1 and Theorem 3]: The functions $\frac1{x^2}-\frac{e^{-x}}{(1-e^{-x})^2}\;and\;\frac1{t}-\frac1{e^t-1}$ are decreasing in (0, ${\infty}$) and the function $\frac{t}{e^{at}-e^{(a-1)t}}$ for a $a{\in}\mathbb{R}\;and\;t\;{\in}\;(0,\;{\infty})$ is logarithmically concave.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.29-34
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• 1993

J. Leray [7] proposed a sufficient condition ofr the solvability of the Cauchy problem on the initial hyperplane x$_{1}$=0 with Cauchy data which are holomorphic with respect to the variables parallel to some analytic subvariety S of the initial hyperplane. Limiting the problem to the case of operators with constant coefficients, A. Kaneko [2] proposed a new sharper sufficient condition. Later we generalized this condition and showed that it is necessary and sufficient for the solvability of the Cauchy problem for the hyperfunction Cauchy data and the distribution Cauchy data which contain variables parallel to S as holomorphic parameters in [5, 6]. In this paper, we extend the results in [6] to the case of operators with variable coefficients and show that it is sufficient for the solvability of the Cauchy problem for the hyperfunction Cauchy data. Our main theorem can be considered as an example of a deep theorem on micro-hyperbolic systems by Kashiwara-Schapira [4] and we give a direct proof based on an elementary sweeping out procedure developed in Kaneko [3].