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Spectrum of a linear operator in a family of banach spaces containing Hilbert space

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Abstract

The mutual relation is established between the spectra of a bounded linear operator acting in a family of Banach spaces. It is assumed in addition that one of the spaces is a Hilbert space and that the operator acting on it is self-adjoint. An example is presented illustrating the properties proved.

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Literature cited

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    S. G. Krein and Yu. I. Petunin, “Scales of Banach spaces,” Usp. Mat. Nauk,13, No. 2, 89–168 (1966).

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    M. G. Krein, “On linear completely continuous operators in function spaces with two norms,” Tr. Inst. Mat. Akad. Nauk Ukr. SSR,9, 104–129 (1947).

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    C. Halberg and A. E. Taylor, Jr., “On the spectra of linked operators,” Pac. J. Math.,6, No. 2, 283–290 (1956).

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Translated from Matematicheskie Zametki, Vol. 22, No. 4, pp. 495–498, October, 1977.

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Shevchik, V.V. Spectrum of a linear operator in a family of banach spaces containing Hilbert space. Mathematical Notes of the Academy of Sciences of the USSR 22, 767–769 (1977). https://doi.org/10.1007/BF01146420

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Keywords

  • Hilbert Space
  • Banach Space
  • Linear Operator
  • Bounded Linear Operator
  • Mutual Relation