• Title, Summary, Keyword: The continuity of the spectrum

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CONTINUITY OF APPROXIMATE POINT SPECTRUM ON THE ALGEBRA B(X)

  • Sanchez-Perales, Salvador;Cruz-Barriguete, Victor A.
    • Communications of the Korean Mathematical Society
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    • v.28 no.3
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    • pp.487-500
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    • 2013
  • In this paper we provide a brief introduction to the continuity of approximate point spectrum on the algebra B(X), using basic properties of Fredholm operators and the SVEP condition. Also, we give an example showing that in general it not holds that if the spectrum is continuous an operator T, then for each ${\lambda}{\in}{\sigma}_{s-F}(T){\setminus}\overline{{\rho}^{\pm}_{s-F}(T)}$ and ${\in}$ > 0, the ball $B({\lambda},{\in})$ contains a component of ${\sigma}_{s-F}(T)$, contrary to what has been announced in [J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity II, Integral Equations Operator Theory 4 (1981), 459-503] page 462.

THE SPECTRAL CONTINUITY OF ESSENTIALLY HYPONORMAL OPERATORS

  • Kim, An-Hyun;Ryu, Eun-Jin
    • Communications of the Korean Mathematical Society
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    • v.29 no.3
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    • pp.401-408
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    • 2014
  • If A is a unital Banach algebra, then the spectrum can be viewed as a function ${\sigma}$ : 𝕬 ${\rightarrow}$ 𝕾, mapping each T ${\in}$ 𝕬 to its spectrum ${\sigma}(T)$, where 𝕾 is the set, equipped with the Hausdorff metric, of all compact subsets of $\mathbb{C}$. This paper is concerned with the continuity of the spectrum ${\sigma}$ via Browder's theorem. It is shown that ${\sigma}$ is continuous when ${\sigma}$ is restricted to the set of essentially hyponormal operators for which Browder's theorem holds, that is, the Weyl spectrum and the Browder spectrum coincide.

WEYL'S THEOREM, TENSOR PRODUCT, FUGLEDE-PUTNAM THEOREM AND CONTINUITY SPECTRUM FOR k-QUASI CLASS An* OPERATO

  • Hoxha, Ilmi;Braha, Naim Latif
    • Journal of the Korean Mathematical Society
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    • v.51 no.5
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    • pp.1089-1104
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    • 2014
  • An operator $T{\in}L(H)$, is said to belong to k-quasi class $A_n^*$ operator if $$T^{*k}({\mid}T^{n+1}{\mid}^{\frac{2}{n+1}}-{\mid}T^*{\mid}^2)T^k{\geq}O$$ for some positive integer n and some positive integer k. First, we will see some properties of this class of operators and prove Weyl's theorem for algebraically k-quasi class $A_n^*$. Second, we consider the tensor product for k-quasi class $A_n^*$, giving a necessary and sufficient condition for $T{\otimes}S$ to be a k-quasi class $A_n^*$, when T and S are both non-zero operators. Then, the existence of a nontrivial hyperinvariant subspace of k-quasi class $A_n^*$ operator will be shown, and it will also be shown that if X is a Hilbert-Schmidt operator, A and $(B^*)^{-1}$ are k-quasi class $A_n^*$ operators such that AX = XB, then $A^*X=XB^*$. Finally, we will prove the spectrum continuity of this class of operators.

ABSOLUTE CONTINUITY OF THE MAGNETIC SCHRÖDINGER OPERATOR WITH PERIODIC POTENTIAL

  • Assel, Rachid
    • Korean Journal of Mathematics
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    • v.26 no.4
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    • pp.601-614
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    • 2018
  • We consider the magnetic $Schr{\ddot{o}}dinger$ operator coupled with two different potentials. One of them is a harmonic oscillator and the other is a periodic potential. We give some periodic potential classes for which the operator has purely absolutely continuous spectrum. We also prove that for strong magnetic field or large coupling constant, there are open gaps in the spectrum and we give a lower bound on their number.

SPECTRAL CONTINUITY OF ESSENTIALLY p-HYPONORMAL OPERATORS

  • Kim, An-Hyun;Kwon, Eun-Young
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.2
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    • pp.389-393
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    • 2006
  • In this paper it is shown that the spectrum ${\sigma}$ is continuous at every p-hyponormal operator when restricted to the set of essentially p-hyponormal operators and moreover ${\sigma}$ is continuous when restricted to the set of compact perturbations of p-hyponormal operators whose spectral pictures have no holes associated with the index zero.