• Title/Summary/Keyword: class A(k) operators

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LINEAR MAPS PRESERVING 𝓐𝓝-OPERATORS

  • Golla, Ramesh;Osaka, Hiroyuki
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.4
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    • pp.831-838
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    • 2020
  • Let H be a complex Hilbert space and T : H → H be a bounded linear operator. Then T is said to be norm attaining if there exists a unit vector x0 ∈ H such that ║Tx0║ = ║T║. If for any closed subspace M of H, the restriction T|M : M → H of T to M is norm attaining, then T is called an absolutely norm attaining operator or 𝓐𝓝-operator. In this note, we discuss linear maps on B(H), which preserve the class of absolutely norm attaining operators on H.

EXISTENCE OF THREE POSITIVE SOLUTIONS OF A CLASS OF BVPS FOR SINGULAR SECOND ORDER DIFFERENTIAL SYSTEMS ON THE WHOLE LINE

  • Liu, Yuji;Yang, Pinghua
    • Journal of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.359-380
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    • 2017
  • This paper is concerned with a kind of boundary value problem for singular second order differential systems with Laplacian operators. Using a multiple fixed point theorem, sufficient conditions to guarantee the existence of at least three positive solutions of this kind of boundary value problem are established. An example is presented to illustrate the main results.

EXISTENCE OF THREE SOLUTIONS FOR A NAVIER BOUNDARY VALUE PROBLEM INVOLVING THE p(x)-BIHARMONIC

  • Yin, Honghui;Liu, Ying
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1817-1826
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    • 2013
  • The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operators with Navier boundary value conditions. The technical approach is mainly based on a three critical points theorem due to Ricceri [11].

SAMPLING THEOREMS ASSOCIATED WITH DIFFERENTIAL OPERATORS WITH FINITE RANK PERTURBATIONS

  • Annaby, Mahmoud H.;El-Haddad, Omar H.;Hassan, Hassan A.
    • Journal of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.969-990
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    • 2016
  • We derive a sampling theorem associated with first order self-adjoint eigenvalue problem with a finite rank perturbation. The class of the sampled integral transforms is of finite Fourier type where the kernel has an additional perturbation.

MULTIPLIERS FOR OPERATOR-VALUED BESSEL SEQUENCES AND GENERALIZED HILBERT-SCHMIDT CLASSES

  • KRISHNA, K. MAHESH;JOHNSON, P. SAM;MOHAPATRA, R.N.
    • Journal of applied mathematics & informatics
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    • v.40 no.1_2
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    • pp.153-171
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    • 2022
  • In 1960, Schatten studied operators of the form $\sum_{n=1}^{{\infty}}\;{\lambda}_n(x_n{\otimes}{\bar{y_n}})$, where {xn}n and {yn}n are orthonormal sequences in a Hilbert space, and {λn}n ∈ ℓ(ℕ). Balazs generalized some of the results of Schatten in 2007. In this paper, we further generalize results of Balazs by studying the operators of the form $\sum_{n=1}^{{\infty}}\;{\lambda}_n(A^*_nx_n{\otimes}{\bar{B^*_ny_n}})$, where {An}n and {Bn}n are operator-valued Bessel sequences, {xn}n and {yn}n are sequences in the Hilbert space such that {║xn║║yn║}n ∈ ℓ(ℕ). We also generalize the class of Hilbert-Schmidt operators studied by Balazs.

New Sufficient Conditions for Starlikeness of Certain Integral Operator

  • Mishra, Akshaya Kumar;Panigrahi, Trailokya
    • Kyungpook Mathematical Journal
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    • v.55 no.1
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    • pp.109-118
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    • 2015
  • In the present paper, a new analytic function valued integral operator is introduced which is defined on n-copies of a subset of the class of normalized analytic functions on the unit disc of the complex plane. This operator, which is denoted here by $\mathfrak{J}^{{\alpha}_i,{\beta}_i}(f_1,{\ldots},f_n)$, unifies and generalizes several integral operators studied earlier. Interesting sufficient conditions are derived for the univalent starlikeness of $\mathfrak{J}^{{\alpha}_i,{\beta}_i}(f_1,{\ldots},f_n)$.

BOUNDEDNESS AND INVERSION PROPERTIES OF CERTAIN CONVOLUTION TRANSFORMS

  • Yakubovich, Semyon-B.
    • Journal of the Korean Mathematical Society
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    • v.40 no.6
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    • pp.999-1014
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    • 2003
  • For a fixed function h we deal with a class of convolution transforms $f\;{\rightarrow}\;f\;*\;h$, where $(f\;*\;h)(x)\;=\frac{1}{2x}\;{\int_{{R_{+}}^2}}^{e^1{\frac{1}{2}}(x\frac{u^2+y^2}{uy}+\frac{yu}{x})}\;f(u)h(y)dudy,\;x\;\in\;R_{+}$ as integral operators $L_p(R_{+};xdx)\;\rightarrow\;L_r(R_{+};xdx),\;p,\;r\;{\geq}\;1$. The Young type inequality is proved. Boundedness properties are investigated. Certain examples of these operators are considered and inversion formulas in $L_2(R_{+};xdx)$ are obtained.

A geometric criterion for the element of the class $A_{1,aleph_0 $(r)

  • Kim, Hae-Gyu;Yang, Young-Oh
    • Journal of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.635-647
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    • 1995
  • Let $H$ denote a separable, infinite dimensional complex Hilbert space and let $L(H)$ denote the algebra of all bounded linear operators on $H$. A dual algebra is a subalgebra of $L(H)$ that contains the identity operator $1_H$ and is closed in the $weak^*$ operator topology on $L(H)$. For $T \in L(H)$, let $A_T$ denote the smallest subalgebra of $L(H)$ that contains T and $1_H$ and is closed in the $weak^*$ operator topology.

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REDUCING SUBSPACES OF A CLASS OF MULTIPLICATION OPERATORS

  • Liu, Bin;Shi, Yanyue
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1443-1455
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    • 2017
  • Let $M_{z^N}(N{\in}{\mathbb{Z}}^d_+)$ be a bounded multiplication operator on a class of Hilbert spaces with orthogonal basis $\{z^n:n{\in}{\mathbb{Z}}^d_+\}$. In this paper, we prove that each reducing subspace of $M_{z^N}$ is the direct sum of some minimal reducing subspaces. For the case that d = 2, we find all the minimal reducing subspaces of $M_{z^N}$ ($N=(N_1,N_2)$, $N_1{\neq}N_2$) on weighted Bergman space $A^2_{\alpha}({\mathbb{B}}_2)$(${\alpha}$ > -1) and Hardy space $H^2({\mathbb{B}}_2)$, and characterize the structure of ${\mathcal{V}}^{\ast}(z^N)$, the commutant algebra of the von Neumann algebra generated by $M_{z^N}$.

ITERATIVE ALGORITHMS FOR A FUZZY SYSTEM OF RANDOM NONLINEAR EQUATIONS IN HILBERT SPACES

  • Salahuddin, Salahuddin
    • Communications of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.333-352
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    • 2017
  • In this research work, by using the random resolvent operator techniques associated with random ($A_t$, ${\eta}_t$, $m_t$)-monotone operators, is to established an existence and convergence theorems for a class of fuzzy system of random nonlinear equations with fuzzy mappings in Hilbert spaces. Our results improve and generalized the corresponding results of the recent works.