• Title/Summary/Keyword: normal operator

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On Normal Products of Selfadjoint Operators

  • Jung, Il Bong;Mortad, Mohammed Hichem;Stochel, Jan
    • Kyungpook Mathematical Journal
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    • v.57 no.3
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    • pp.457-471
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    • 2017
  • A necessary and sufficient condition for the product AB of a selfadjoint operator A and a bounded selfadjoint operator B to be normal is given. Various properties of the factors of the unitary polar decompositions of A and B are obtained in the case when the product AB is normal. A block operator model for pairs (A, B) of selfadjoint operators such that B is bounded and AB is normal is established. The case when both operators A and B are bounded is discussed. In addition, the example due to Rehder is reexamined from this point of view.

NORMAL, COHYPONORMAL AND NORMALOID WEIGHTED COMPOSITION OPERATORS ON THE HARDY AND WEIGHTED BERGMAN SPACES

  • Fatehi, Mahsa;Shaabani, Mahmood Haji
    • Journal of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.599-612
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    • 2017
  • If ${\psi}$ is analytic on the open unit disk $\mathbb{D}$ and ${\varphi}$ is an analytic self-map of $\mathbb{D}$, the weighted composition operator $C_{{\psi},{\varphi}}$ is defined by $C_{{\psi},{\varphi}}f(z)={\psi}(z)f({\varphi}(z))$, when f is analytic on $\mathbb{D}$. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces $H^2({\beta})$, we prove that if $C_{{\psi},{\varphi}}$ is cohyponormal on $H^2({\beta})$, then ${\psi}$ never vanishes on $\mathbb{D}$ and ${\varphi}$ is univalent, when ${\psi}{\not\equiv}0$ and ${\varphi}$ is not a constant function. Moreover, for ${\psi}=K_a$, where |a| < 1, we investigate normal, cohyponormal and hyponormal weighted composition operators $C_{{\psi},{\varphi}}$. After that, for ${\varphi}$ which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators $C_{{\psi},{\varphi}}$, when ${\psi}{\not\equiv}0$ and ${\psi}$ is analytic on $\bar{\mathbb{D}}$. Finally, we find all normal weighted composition operators which are bounded below.

HOPF HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH LIE PARALLEL NORMAL JACOBI OPERATOR

  • Jeong, Im-Soon;Lee, Hyun-Jin;Suh, Young-Jin
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.2
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    • pp.427-444
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    • 2011
  • In this paper we give some non-existence theorems for Hopf hypersurfaces in the complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ with Lie parallel normal Jacobi operator $\bar{R}_N$ and totally geodesic D and $D^{\bot}$ components of the Reeb flow.

REAL HYPERSURFACES OF TYPE A IN COMPLEX TWO-PLANE GRASSMANNIANS RELATED TO THE NORMAL JACOBI OPERATOR

  • Jeong, Im-Soon;Suh, Young-Jin;Tripathi, Mukut Mani
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.2
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    • pp.423-434
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    • 2012
  • In this paper we give a characterization of real hypersurfaces of type (A) in a complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ which is a tube over a totally geodesic $G_2(\mathbb{C}^{m+1})$ in $G_2(\mathbb{C}^{m+2})$, in terms of two commuting conditions related to the normal Jacobi operator and the shape operator.

MIXED RADIAL-ANGULAR INTEGRABILITIES FOR HARDY TYPE OPERATORS

  • Ronghui Liu ;Shuangping Tao
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.5
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    • pp.1409-1425
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    • 2023
  • In this paper, we are devoted to studying the mixed radial-angular integrabilities for Hardy type operators. As an application, the upper and lower bounds are obtained for the fractional Hardy operator. In addition, we also establish the sharp weak-type estimate for the fractional Hardy operator.

Real Hypersurfaces with Invariant Normal Jacobi Operator in the Complex Hyperbolic Quadric

  • Jeong, Imsoon;Kim, Gyu Jong
    • Kyungpook Mathematical Journal
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    • v.60 no.3
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    • pp.551-570
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    • 2020
  • We introduce the notion of Lie invariant normal Jacobi operators for real hypersurfaces in the complex hyperbolic quadric Qm∗ = SOom,2/SOmSO2. The invariant normal Jacobi operator implies that the unit normal vector field N becomes 𝕬-principal or 𝕬-isotropic. Then in each case, we give a complete classification of real hypersurfaces in Qm∗ = SOom,2/SOmSO2 with Lie invariant normal Jacobi operators.

NORMAL INTERPOLATION ON AX=Y AND Ax=y IN A TRIDIAGONAL ALGEBRA $ALG\mathcal{L}$

  • Kang, Joo-Ho
    • Journal of applied mathematics & informatics
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    • v.24 no.1_2
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    • pp.535-539
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    • 2007
  • Given operators X and Y acting on a separable complex Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX=Y. In this article, we show the following: Let $Alg\mathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $\mathcal{H}$ and let $X=(x_{ij})\;and\;Y=(y_{ij})$ be operators in $\mathcal{H}$. Then the following are equivalent: (1) There exists a normal operator $A=(a_{ij})\;in\;Alg\mathcal{L}$ such that AX=Y. (2) There is a bounded sequence $\{\alpha_n\}\;in\;\mathbb{C}$ such that $y_{ij}=\alpha_jx_{ij}\;for\;i,\;j\;{\in}\;\mathbb{N}$. Given vectors x and y in a separable complex Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that Ax=y. We show the following: Let $Alg\mathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $\mathcal{H}$ and let $x=(x_i)\;and\;y=(y_i)$ be vectors in $\mathcal{H}$. Then the following are equivalent: (1) There exists a normal operator $A=(a_{ij})\;in\;Alg\mathcal{L}$ such that Ax=y. (2) There is a bounded sequence $\{\alpha_n\}$ in $\mathbb{C}$ such that $y_i=\alpha_ix_i\;for\;i{\in}\mathbb{N}$.