• Title, Summary, Keyword: Hilbert space

### AN EXISTENCE OF LINEAR SYSTEMS WITH GIVEN TRANSFER FUNCTION

• Yang, Meehyea
• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.99-107
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• 1993
• A vector space K with scalar product <.,.> is called a Krein space if it can be decomposed as a northogonal sum of a Hilbert space and an anti-space of a Hilbert space. The space K induces a Hilbert space $K_{J}$ in the inner product <.,.> $K_{J}$=<.,.>K, where $J^{2}$=I. the eigenspaces of J are denoted by $K^{+}$$_{J}, which is a Hilbert space and K^{-}$$_{J}$, which is an anti-space of a Hilbert space. Then the Krein space K is the orthogonal sum of $K^{+}$$_{J} and K^{-}$$_{J}$. Such a decomposition of K is called a fundamental decomposition. In general, fundamental decompositions are not unique. The norm of the Hilbert space depends on the choice of a fundamental decomposion, but such norms are equivalent. The topology generated by these norms is called the strong or Mackey topology of K. It is used to define all topological notions on the Krein space K with respect to this topology. The Pontryagin index of a Krein space is the dimension of the antispace of a Hilbert space in any such decomposition. the dimension does not depend on the choice of orthogonal decomposition. A Krein space is called a Pontryagin space if it has finite Pontryagin index.dex.yagin index.dex.

### A STRONG LAW OF LARGE NUMBERS FOR AANA RANDOM VARIABLES IN A HILBERT SPACE AND ITS APPLICATION

• Ko, Mi-Hwa
• Honam Mathematical Journal
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• v.32 no.1
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• pp.91-99
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• 2010
• In this paper we introduce the concept of asymptotically almost negatively associated random variables in a Hilbert space and obtain the strong law of large numbers for a strictly stationary asymptotically almost negatively associated sequence of H-valued random variables with zero means and finite second moments. As an application we prove a strong law of large numbers for a linear process generated by asymptotically almost negatively random variables in a Hilbert space with this result.

### ON THE THREE OPERATOR SPACE STRUCTURES OF HILBERT SPACES

• Shin, Dong-Yun
• Communications of the Korean Mathematical Society
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• v.11 no.4
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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.

### A Central Limit Theorem for the Linear Process in a Hilbert Space under Negative Association

• Ko, Mi-Hwa
• Communications for Statistical Applications and Methods
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• v.16 no.4
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• pp.687-696
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• 2009
• We prove a central limit theorem for the negatively associated random variables in a Hilbert space and extend this result to the linear process generated by negatively associated random variables in a Hilbert space. Our result implies an extension of the central limit theorem for the linear process in a real space under negative association to a simplest case of infinite dimensional Hilbert space.

### A Difference of Two Composition Operators on L2 and H2

• Nakazi, Takahiko
• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.125-130
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• 2016
• A finite rank difference of two composition operators is studied on a Hilbert Lebesgue space or a Hilbert Hardy space.

### STOCHASTIC INTEGRAL OF PROCESSES TAKING VALUES OF GENERALIZED OPERATORS

• CHOI, BYOUNG JIN;CHOI, JIN PIL;JI, UN CIG
• Journal of applied mathematics & informatics
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• v.34 no.1_2
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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.

### FURTHER RESULTS INVOLVING THE HILBERT SPACE L2a,b[0, T]

• Choi, Jae Gil;Skoug, David
• The Pure and Applied Mathematics
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• v.27 no.1
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• pp.1-11
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• 2020
• In this paper we determine conditions which a function a(t) must satisfy to insure that the function a'(t) is an element of the separable Hilbert space L2a,b[0, T]. We then proceed to illustrate our results with several pertinent examples and counter-examples.

### THE GENERALIZED INVERSES A(1,2)T,S OF THE ADJOINTABLE OPERATORS ON THE HILBERT C^*-MODULES

• Xu, Qingxiang;Zhang, Xiaobo
• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.363-372
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• 2010
• In this paper, we introduce and study the generalized inverse $A^{(1,2)}_{T,S}$ with the prescribed range T and null space S of an adjointable operator A from one Hilbert $C^*$-module to another, and get some analogous results known for finite matrices over the complex field or associated rings, and the Hilbert space operators.

### CHARACTERIZATION OF THE HILBERT BALL BY ITS AUTOMORPHISMS

• Kim, Kang-Tae;Ma, Daowei
• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.503-516
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• 2003
• We show in this paper that every domain in a separable Hilbert space, say H, which has a $C^2$ smooth strongly pseudoconvex boundary point at which an automorphism orbit accumulates is biholomorphic to the unit ball of H. This is the complete generalization of the Wong-Rosay theorem to a separable Hilbert space of infinite dimension. Our work here is an improvement from the preceding work of Kim/Krantz [10] and subsequent improvement of Byun/Gaussier/Kim [3] in the infinite dimensions.

### THE WEAK LAWS OF LARGE NUMBERS FOR SUMS OF ASYMPTOTICALLY ALMOST NEGATIVELY ASSOCIATED RANDOM VECTORS IN HILBERT SPACES

• Kim, Hyun-Chull
• Journal of the Chungcheong Mathematical Society
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• v.32 no.3
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• pp.327-336
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• 2019
• In this paper, the weak laws of large numbers for sums of asymptotically almost negatively associated random vectors in Hilbert spaces are investigated. Some results in Hien and Thanh ([3]) are generalized to asymptotically almost negatively random vectors in Hilbert space.