• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.269-280
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• 1998

Elementary properties of the sequence space l(s, t) are studied with applications to Hardy space theory.

• Bulletin of the Korean Mathematical Society
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• v.29 no.2
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• pp.295-300
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• 1992

In [5] it was shown that if T .mem. L(X) is regular on a Banach space X, with finite dimensional intersection T$^{-1}$ (0).cap.T(X) and if S .mem. L(X) is invertible, commute with T and has sufficiently small norm then T - S in upper semi-Fredholm, and hence essentially one-one, in the sense that the null space of T - S is finite dimensional ([4] Theorem 2; [5] Theorem 2). In this note we extend this result to incomplete normed space.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1117-1127
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• 2019

Let ${\mathcal{L}}_2=(-{\Delta})^2+V^2$ be the $Schr{\ddot{o}}dinger$ type operator, where nonnegative potential V belongs to the reverse $H{\ddot{o}}lder$ class $RH_s$, s > n/2. In this paper, we consider the operator $T_{{\alpha},{\beta}}=V^{2{\alpha}}{\mathcal{L}}^{-{\beta}}_2$ and its conjugate $T^*_{{\alpha},{\beta}}$, where $0<{\alpha}{\leq}{\beta}{\leq}1$. We establish the $(L^p,\;L^q)$-boundedness of operator $T_{{\alpha},{\beta}}$ and $T^*_{{\alpha},{\beta}}$, respectively, we also show that $T_{{\alpha},{\beta}}$ is bounded from Hardy type space $H^1_{L_2}({\mathbb{R}}^n)$ into $L^{p_2}({\mathbb{R}}^n)$ and $T^*_{{\alpha},{\beta}}$ is bounded from $L^{p_1}({\mathbb{R}}^n)$ into BMO type space $BMO_{{\mathcal{L}}1}({\mathbb{R}}^n)$, where $p_1={\frac{n}{4({\beta}-{\alpha})}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})}}$.

• Bulletin of the Korean Mathematical Society
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• v.34 no.1
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• pp.19-28
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• 1997

Let $H$ and $K$ be separable, complex Hilbert spaces and $L(H, K)$ denote the space of all linear, bounded operators from $H$ to $K$. If $H = K$, we write $L(H)$ in place of $L(H, K)$. An operator $T$ in $L(H)$ is called hyponormal if $TT^* \leq T^*T$, or equivalently, if $\left\$\mid$ T^*h \right\$\mid$ \leq \left\$\mid$ Th \right\$\mid$$ for each h in $H$. In [Pu], M. Putinar constructed a universal functional model for hyponormal operators.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.709-723
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• 2016

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t] and define a random vector $Z_n:C[0,t]{\rightarrow}{\mathbb{R}}^n$ by $Zn(x)=(\int_{0}^{t_1}h(s)dx(s),{\cdots},\int_{0}^{t_n}h(s)dx(s))$, where 0 < $t_1$ < ${\cdots}$ < $t_n$ < t is a partition of [0, t] and $h{\in}L_2[0,t]$ with $h{\neq}0$ a.e. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on C[0, t] with the conditioning function $Z_n$ and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $F(x)=f(\int_{0}^{t}e(s)dx(s))$ for $x{\in}C[0,t]$, where $f{\in}L_p(\mathbb{R})(1{\leq}p{\leq}{\infty})$ and e is a unit element in $L_2[0,t]$. Finally we express the generalized analytic conditional Feynman integral of F as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of e and the conditioning function $Z_n$ does not contain the present positions of the generalized Wiener paths.

• Journal of the Korean Mathematical Society
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• v.44 no.4
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• pp.1025-1050
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• 2007

Let $X_k(x)=({\int}^T_o{\alpha}_1(s)dx(s),...,{\int}^T_o{\alpha}_k(s)dx(s))\;and\;X_{\tau}(x)=(x(t_1),...,x(t_k))$ on the classical Wiener space, where ${{\alpha}_1,...,{\alpha}_k}$ is an orthonormal subset of $L_2$ [0, T] and ${\tau}:0 is a partition of [0, T]. In this paper, we establish a change of scale formula for conditional Wiener integrals $E[G_{\gamma}|X_k]$ of functions on classical Wiener space having the form $$G_{\gamma}(x)=F(x){\Psi}({\int}^T_ov_1(s)dx(s),...,{\int}^T_o\;v_{\gamma}(s)dx(s))$$, for $F{\in}S\;and\;{\Psi}={\psi}+{\phi}({\psi}{\in}L_p(\mathbb{R}^{\gamma}),\;{\phi}{\in}\hat{M}(\mathbb{R}^{\gamma}))$, which need not be bounded or continuous. Here S is a Banach algebra on classical Wiener space and $\hat{M}(\mathbb{R}^{\gamma})$ is the space of Fourier transforms of measures of bounded variation over $\mathbb{R}^{\gamma}$. As results of the formula, we derive a change of scale formula for the conditional Wiener integrals $E[G_{\gamma}|X_{\tau}]\;and\;E[F|X_{\tau}]$. Finally, we show that the analytic Feynman integral of F can be expressed as a limit of a change of scale transformation of the conditional Wiener integral of F using an inversion formula which changes the conditional Wiener integral of F to an ordinary Wiener integral of F, and then we obtain another type of change of scale formula for Wiener integrals of F.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.119-126
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• 1989

Throughout this paper, X will always denote a Banach space over the complex numbers C, and L(X) will denote the Banach algebra of all continuous linear operators on X. Operator will always mean continuous linear operator. An n-tuple of operators T$_{1}$,..,T$_{n}$ on X will be denoted by over ^ T=(T$_{1}$,..,T$_{n}$ ). Let L$^{n}$ (X) be the set of all n-tuples of operators on X. X' will denote the dual space of X, S(X) its unit sphere and .PI.(X) the subset of X*X' defined by .PI.(X)={(x,f).mem.X*X': ∥x∥=∥f∥=f(x)=1}.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.499-507
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• 2010

In this note, we study the local spectral properties of semi-shifts. If $T\;{\in}\;L(X)$ is a semi-shift on a complex Banach space X, then T is admissible. We also prove that if $T\;{\in}\;L(X)$ is subadmissible, then $X_T(F)\;=\;E_T(F)$ for all closed $F\;{\subseteq}\;\mathbb{C}$. In particular, every subscalar operator on a Banach space is admissible.

• Honam Mathematical Journal
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• v.35 no.4
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• pp.707-716
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• 2013

The present paper consists of two parts. Since the recent paper [4] proved that an Alexandroff $T_0$-space is a semi-$T_{\frac{1}{2}}$-space, the first part studies semi-open and semi-closed structures of the Khalimsky nD space. The second one focuses on the study of a relation between the LS-property of ($SC^{n_1,l_1}_{k_1}{\times}SC^{n_2,l_2}_{k_2}$, k) relative to the simple closed $k_i$-curves $SC^{n_i,l_i}_{k_i}$, $i{\in}\{1,2\}$ and its normal k-adjacency. In addition, the present paper points out that the main theorems of Boxer and Karaca's paper [3] such as Theorems 4.4 and 4.7 of [3] cannot be new assertions. Indeed, instead they should be attributed to Theorems 4.3 and 4.5, and Example 4.6 of [10].

• Communications of the Korean Mathematical Society
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• v.19 no.1
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• pp.93-111
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• 2004

In this paper, we use a generalized Brownian motion process to define a generalized Feynman integral and a generalized Fourier-Feynman transform. We also define the concepts of the multiple Lp analytic generalized Fourier-Feynman transform and the generalized convolution product of functional on function space $C_{a,\;b}[0,\;T]$. We then verify the existence of the multiple $L_{p}$ analytic generalized Fourier-Feynman transform for functional on function space that belong to a Banach algebra $S({L_{a,\;b}}^{2}[0, T])$. Finally we establish some relationships between the multiple $L_{p}$ analytic generalized Fourier-Feynman transform and the generalized convolution product for functionals in $S({L_{a,\;b}}^{2}[0, T])$.