• Title, Summary, Keyword: Fredholm operator

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THE JUMP OF A SEMI-FREDHOLM OPERATOR

  • Lee, Dong-Hak;Lee, Woo-Young
    • Communications of the Korean Mathematical Society
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    • v.9 no.3
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    • pp.593-598
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    • 1994
  • In this note we give some results on the jump (due to Kato [5] and West [7]) of a semi-Fredholm operator. Throughout this note, suppose X is an Banach space and write L(X) for the set of all bounded linear operators on X. A operator $T \in L(x)$ is called upper semi-Fredholm if it has closed range with finite dimensional null space, and lower semi-Fredholm if it has closed range with its range of finite co-dimension. It T is either upper or lower semi-Fredholm we shall call it semi-Fredholm and Fredholm it is both. The index of a (semi-) Fredholm operator T is given by $$ index(T) = n(T) = d(T),$$ where $n(T) = dim T^{-1}(0)$ and d(T) = codim T(X).

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CHARACTERIZATION OF RELATIVELY DEMICOMPACT OPERATORS BY MEANS OF MEASURES OF NONCOMPACTNESS

  • Jeribi, Aref;Krichen, Bilel;Salhi, Makrem
    • Journal of the Korean Mathematical Society
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    • v.55 no.4
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    • pp.877-895
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    • 2018
  • In this paper, we show that an unbounded $S_0$-demicompact linear operator T with respect to a bounded linear operator $S_0$, acting on a Banach space, can be characterized by the Kuratowskii measure of noncompactness. Moreover, some other quantities related to this measure provide sufficient conditions to the operator T to be $S_0$-demicompact. The obtained results are used to discuss the connection with Fredholm and upper Semi-Fredholm operators.

SPECTRA ORIGINATED FROM FREDHOLM THEORY AND BROWDER'S THEOREM

  • Amouch, Mohamed;Karmouni, Mohammed;Tajmouati, Abdelaziz
    • Communications of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.853-869
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    • 2018
  • We give a new characterization of Browder's theorem through equality between the pseudo B-Weyl spectrum and the generalized Drazin spectrum. Also, we will give conditions under which pseudo B-Fredholm and pseudo B-Weyl spectrum introduced in [9] and [25] become stable under commuting Riesz perturbations.

ZERO BASED INVARIANT SUBSPACES AND FRINGE OPERATORS OVER THE BIDISK

  • Izuchi, Kei Ji;Izuchi, Kou Hei;Izuchi, Yuko
    • Journal of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.847-868
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    • 2016
  • Let M be an invariant subspace of $H^2$ over the bidisk. Associated with M, we have the fringe operator $F^M_z$ on $M{\ominus}{\omega}M$. It is studied the Fredholmness of $F^M_z$ for (generalized) zero based invariant subspaces M. Also ker $F^M_z$ and ker $(F^M_z)^*$ are described.

POLYNOMIALLY DEMICOMPACT OPERATORS AND SPECTRAL THEORY FOR OPERATOR MATRICES INVOLVING DEMICOMPACTNESS CLASSES

  • Brahim, Fatma Ben;Jeribi, Aref;Krichen, Bilel
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.5
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    • pp.1351-1370
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    • 2018
  • In the first part of this paper we show that, under some conditions, a polynomially demicompact operator can be demicompact. An example involving the Caputo fractional derivative of order ${\alpha}$ is provided. Furthermore, we give a refinement of the left and the right Weyl essential spectra of a closed linear operator involving the class of demicompact ones. In the second part of this work we provide some sufficient conditions on the inputs of a closable block operator matrix, with domain consisting of vectors which satisfy certain conditions, to ensure the demicompactness of its closure. Moreover, we apply the obtained results to determine the essential spectra of this operator.