• Title/Summary/Keyword: compact operator

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Every Operator Almost Commutes with a Compact Operator

  • Jung, Il Bong;Ko, Eungil;Pearcy, Carl
    • Kyungpook Mathematical Journal
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    • v.47 no.2
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    • pp.221-226
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    • 2007
  • In this note we set forth three possible definitions of the property of "almost commuting with a compact operator" and discuss an old result of W. Arveson that says that every operator on Hilbert space has the weakest of the three properties. Finally, we discuss some recent progress on the hyperinvariant subspace problem (see the bibliography), and relate it to the concept of almost commuting with a compact operator.

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SELF-ADJOINT CYCLICALLY COMPACT OPERATORS AND ITS APPLICATION

  • Kudaybergenov, Karimbergen;Mukhamedov, Farrukh
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.679-686
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    • 2017
  • The present paper is devoted to self-adjoint cyclically compact operators on Hilbert-Kaplansky module over a ring of bounded measurable functions. The spectral theorem for such a class of operators is given. We use more simple and constructive method, which allowed to apply this result to compact operators relative to von Neumann algebras. Namely, a general form of compact operators relative to a type I von Neumann algebra is given.

SOLVABILITY FOR THE PARABOLIC PROBLEM WITH JUMPING NONLINEARITY CROSSING NO EIGENVALUES

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.16 no.4
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    • pp.545-551
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    • 2008
  • We investigate the multiple solutions for a parabolic boundary value problem with jumping nonlinearity crossing no eigenvalues. We show the existence of the unique solution of the parabolic problem with Dirichlet boundary condition and periodic condition when jumping nonlinearity does not cross eigenvalues of the Laplace operator $-{\Delta}$. We prove this result by investigating the Lipschitz constant of the inverse compact operator of $D_t-{\Delta}$ and applying the contraction mapping principle.

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COMPACT INTERPOLATION ON AX = Y IN ALG𝓛

  • Kang, Joo Ho
    • Journal of applied mathematics & informatics
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    • v.32 no.3_4
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    • pp.441-446
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    • 2014
  • In this paper the following is proved: Let $\mathcal{L}$ be a subspace lattice on a Hilbert space $\mathcal{H}$ and X and Y be operators acting on $\mathcal{H}$. Then there exists a compact operator A in $Alg\mathcal{L}$ such that AX = Y if and only if ${\sup}\{\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}\;:\;f{\in}\mathcal{H},\;E{\in}\mathcal{L}\}$ = K < ${\infty}$ and Y is compact. Moreover, if the necessary condition holds, then we may choose an operator A such that AX = Y and ${\parallel}A{\parallel}=K$.

SPECTRA OF THE IMAGES UNDER THE FAITHFUL $^*$-REPRESENTATION OF L(H) ON K

  • Cha, Hyung-Koo
    • Bulletin of the Korean Mathematical Society
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    • v.22 no.1
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    • pp.23-29
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    • 1985
  • Let H be an arbitrary complex Hilbert space. We constructed an extension K of H by means of weakly convergent sequences in H and the Banach limit. Let .phi. be the faithful *-representation of L(H) on K. In this note, we investigated the relations between spectra of T in L(H) and .phi.(T) in L(K) and we obtained the following results: 1) If T is a compact operator on H, then .phi.(T) is also a compact operator on K (Proposition 6), 2) .sigma.$_{l}$ (.phi.(T)).contnd..sigma.$_{l}$ (T) for any operator T.mem.L(H) (Corollary 10), 3) For every operator T.mem.L(H), .sigma.$_{ap}$ (.phi.(T))=.sigma.$_{ap}$ (T))=.sigma.$_{ap}$ (T)=.sigma.$_{p}$(.phi.(T)) (Lemma 12, 13) and .sigma.$_{c}$(.phi.(T))=.sigma.(Theorem 15).15).

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A NOTE ON APPROXIMATE SIMILARITY

  • Hadwin, Don
    • Journal of the Korean Mathematical Society
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    • v.38 no.6
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    • pp.1157-1166
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    • 2001
  • This paper answers some old questions about approximate similarity and raises new ones. We provide positive evidence and a technique for finding negative evidence on the question of whether approximate similarity is the equivalence relation generated by approximate equivalence and similarity.

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PROXIMINALITY OF CERTAIN SPACES OF COMPACT OPERATORS

  • Cho, Chong-Man;Roh, Woo-Suk
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.65-69
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    • 2001
  • For any closed subspace X of $\ell_p, \; 1<\kappa<\infty$, K(X) is proximinal in L(X), and if X is a Banach space with an unconditional shrinking basis, then K(X, c$_0$) is proximinal in L(X,$ \ell_\infty$).

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GENERALIZED BROWDER, WEYL SPECTRA AND THE POLAROID PROPERTY UNDER COMPACT PERTURBATIONS

  • Duggal, Bhaggy P.;Kim, In Hyoun
    • Journal of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.281-302
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    • 2017
  • For a Banach space operator $A{\in}B(\mathcal{X})$, let ${\sigma}(A)$, ${\sigma}_a(A)$, ${\sigma}_w(A)$ and ${\sigma}_{aw}(A)$ denote, respectively, its spectrum, approximate point spectrum, Weyl spectrum and approximate Weyl spectrum. The operator A is polaroid (resp., left polaroid), if the points $iso{\sigma}(A)$ (resp., $iso{\sigma}_a(A)$) are poles (resp., left poles) of the resolvent of A. Perturbation by compact operators preserves neither SVEP, the single-valued extension property, nor the polaroid or left polaroid properties. Given an $A{\in}B(\mathcal{X})$, we prove that a sufficient condition for: (i) A+K to have SVEP on the complement of ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) for every compact operator $K{\in}B(\mathcal{X})$ is that ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) has no holes; (ii) A + K to be polaroid (resp., left polaroid) for every compact operator $K{\in}B(\mathcal{X})$ is that iso${\sigma}_w(A)$ = ∅ (resp., $iso{\sigma}_{aw}(A)$ = ∅). It is seen that these conditions are also necessary in the case in which the Banach space $\mathcal{X}$ is a Hilbert space.

ON THE RELATION BETWEEN COMPACTNESS AND STRUCTURE OF CERTAIN OPERATORS ON SPACES OF ANALYTIC FUNCTIONS

  • ROBATI, B. KHANI
    • Honam Mathematical Journal
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    • v.23 no.1
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    • pp.29-39
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    • 2001
  • Let $\mathcal{B}$ be a Banach space of analytic functions defined on the open unit disk. Assume S is a bounded operator defined on $\mathcal{B}$ such that S is in the commutant of $M_zn$ or $SM_zn=-M_znS$ for some positive integer n. We give necessary and sufficient condition between compactness of $SM_z+cM_zS$ where c = 1, -1, i, -i, and the structure of S. Also we characterize the commutant of $M_zn$ for some positive integer n.

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