• Korean Journal of Mathematics
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• 제17권2호
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• pp.219-231
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• 2009

We show the existence of at least four solutions for the perturbed parabolic equation with Dirichlet boundary condition and periodic condition when the nonlinear part cross two eigenvalues of the eigenvalue problem of the Laplace operator with boundary condition. We obtain this result by using the Leray-Schauder degree theory, the finite dimensional reduction method and the geometry of the mapping. The main point is that we restrict ourselves to the real Hilbert space instead of the complex space.

• Kyungpook Mathematical Journal
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• 제56권1호
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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.

• 대한수학회지
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• 제44권4호
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• pp.889-902
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• 2007

An existence theorem for a new class of parametric generalized mixed implicit quasi-variational inclusion problems is established in Hilbert spaces. Further, we study the behavior and sensitivity analysis of the solution set in this class of parametric variational inclusion problems.

• Kyungpook Mathematical Journal
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• 제58권4호
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• pp.637-650
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• 2018

In this paper first we show properties of isosymmetric operators given by M. Stankus [13]. Next we introduce an [m, C]-symmetric operator T on a complex Hilbert space H. We investigate properties of the spectrum of an [m, C]-symmetric operator and prove that if T is an [m, C]-symmetric operator and Q is an n-nilpotent operator, respectively, then T + Q is an [m + 2n - 2, C]-symmetric operator. Finally, we show that if T is [m, C]-symmetric and S is [n, D]-symmetric, then $T{\otimes}S$ is [m + n - 1, $C{\otimes}D$]-symmetric.

• 대한수학회논문집
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• 제19권4호
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• pp.661-681
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• 2004

We research properties of the space of measurable functions square integrable with weight exp$2\nu $\mid$x$\mid$$, and those of the space of Fourier hyperfunctions. Also we show that the several embedding theorems hold true, and that the Fourier-Lapace operator is an isomorphism of the space of strongly decreasing Fourier hyperfunctions onto the space of analytic functions extended to any strip in $C^n$ which are estimated with the aid of a special exponential function exp($\mu$｜x｜).

• 대한수학회지
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• 제17권1호
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• pp.123-129
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• 1980

• Korean Journal of Mathematics
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• 제19권1호
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• pp.101-109
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• 2011

We get a theorem which shows that there exist at least two or three nontrivial weak solutions for the nonlinear parabolic boundary value problem with the variable coefficient jumping nonlinearity. We prove this theorem by restricting ourselves to the real Hilbert space. We obtain this result by approaching the topological method. We use the Leray-Schauder degree theory on the real Hilbert space.

• 호남수학학술지
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• 제36권3호
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• pp.679-688
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• 2014

The notion of asymptotically negative dependence for collection of random variables is generalized to a Hilbert space and the almost sure convergence for these H-valued random variables is obtained. The result is also applied to a linear process generated by H-valued asymptotically negatively dependent random variables.

• Korean Journal of Mathematics
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• 제31권1호
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• pp.109-120
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• 2023

The Berezin symbol à of an operator A on the reproducing kernel Hilbert space 𝓗 (Ω) over some set Ω with the reproducing kernel k_λ is defined by $${\tilde{A}}(\lambda)=\,\;{\lambda}{\in}{\Omega}$$. The Berezin number of an operator A is defined by $$ber(A):=\sup_{{\lambda}{\in}{\Omega}}{\mid}{\tilde{A}}({\lambda}){\mid}$$. We study some problems of operator theory by using this bounded function Ã, including estimates for Berezin numbers of some operators, including truncated Toeplitz operators. We also prove an operator analog of some Young inequality and use it in proving of some inequalities for Berezin number of operators including the inequality ber (AB) ≤ ber (A) ber (B), for some operators A and B on 𝓗 (Ω). Moreover, we give in terms of the Berezin number a necessary condition for hyponormality of some operators.

• 대한수학회논문집
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• 제38권3호
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• pp.821-835
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• 2023

Hankel operators and their variants have abundant applications in numerous fields. For a non-zero complex number r, the r-Hankel operators on a Hilbert space 𝓗 define a class of one such variant. This article introduces and explores some properties of two other variants of Hankel operators namely k^th-order (C, r)-Hankel operators and k^th-order (R, r)-Hankel operators (k ≥ 2) which are closely related to r-Hankel operators in such a way that a k^th-order (C, r)-Hankel matrix is formed from r^k-Hankel matrix on deleting every consecutive (k - 1) columns after the first column and a k^th-order (R, r^k)-Hankel matrix is formed from r-Hankel matrix if after the first column, every consecutive (k - 1) columns are deleted. For |r| ≠ 1, the characterizations for the boundedness of these operators are also completely investigated. Finally, an appropriate approach is also presented to extend these matrices to two-way infinite matrices.