• 대한수학회논문집
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• 제9권4호
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• pp.847-852
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• 1994

Let $H$ be a separable, infinite dimensional, complex Hilbert space and let $L(H)$ be the algebra of all bounded linear operator 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 $L(H)$.

• 대한수학회논문집
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• 제11권4호
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• pp.983-996
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• 1996

In this paper, we show that $\Vert \xi \Vert_r = \Vert \sum_{i \in I}x_i x^*_i \Vert^{\frac{1}{2}}, \Vert \xi \Vert_c = \Vert \sum_{i \in I}x^*_ix_i \Vert^{\frac{1}{2}}$ for $\xi = \sum_{i \in I}x_i e_i$ in $M_n(H)$, that subspaces as Hilbert spaces are subspaces as column and row Hilbert spaces, and that the standard dual of column (resp., row) Hilbert spaces is the row (resp., column) Hilbert spaces differently from [1,6]. We define operator Hilbert spaces differently from [10], show that our definition of operator Hilbert spaces is the same as that in [10], show that subspaces as Hilbert spaces are subspaces as operator Hilbert spaces, and for a Hilbert space H we give a matrix norm which is not an operator space norm on H.

• 대한수학회논문집
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• 제34권1호
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• pp.267-277
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• 2019

In this study, we define the dual surfaces by z = f(u) + g(v) and also classify these surfaces in ${\mathbb{I}}{\frac{1}{3}}$ satisfying some algebraic equations in terms of the coordinate functions and the Laplace operators according to fundamental forms of the surface.

• 대한수학회지
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• 제32권4호
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• pp.775-784
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• 1995

Let $H$ be a separable, infnite dimensional, complex Hilbert space and let $L(H)$ denote the algebra of all bounded linear operators on $H$.

• 대한수학회보
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• 제26권2호
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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}.

• 대한수학회논문집
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• 제9권2호
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• pp.317-325
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• 1994

Suppose X is a complex Banach space and write B(X) for the Banach algebra of bounded linear operators on X, X＊ for the dual space of X, and T＊$\in$ B(X＊) for the dual operator of T. For T $\in$ B(X) write a(T) = dim T$^{-1}$ (0) and $\beta$(T) = codim T(X).(omitted)

• 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.

• Journal of applied mathematics & informatics
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• 제34권1_2호
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• pp.167-178
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• 2016

In this paper, we study the stochastic integral of processes taking values of generalized operators based on a triple E ⊂ H ⊂ E^∗, where H is a Hilbert space, E is a countable Hilbert space and E^∗ is the strong dual space of E. For our purpose, we study E-valued Wiener processes and then introduce the stochastic integral of L(E, F^∗)-valued process with respect to an E-valued Wiener process, where F^∗ is the strong dual space of another countable Hilbert space F.

• 대한수학회보
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• 제46권2호
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• pp.263-279
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• 2009

On the setting of the unit ball of ${\mathbb{R}}^n$, we consider a Banach space of harmonic functions motivated by the atomic decomposition in the sense of Coifman and Rochberg [5]. First we identify its dual (resp. predual) space with certain harmonic function space of (resp. vanishing) logarithmic growth. Then we describe these spaces in terms of boundedness and compactness of certain Toeplitz operators.

• 대한수학회지
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• 제49권3호
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• pp.571-583
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• 2012

Let $G$ be a metrizable, ${\sigma}$-compact locally compact abelian group with a compact open subgroup. In this paper we define the Gramian and the dual Gramian operators for shift invariant subspaces of $L^2(G)$ and we use them to characterize shift generated dual frames for shift in- variant spaces, which forms a frame for a subspace of $L^2(G)$. We present necessary and sufficient conditions for which standard dual is a unique SG-dual frame of type I and type II.