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An inequality concerning the smallest disc that contains the spectrum of an operator

Author: Gerhard Garske
Journal: Proc. Amer. Math. Soc. 78 (1980), 529-532
MSC: Primary 47A10
MathSciNet review: 556626
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Abstract: Any bounded linear operator T on a Hilbert space satisfies the inequality $ {\sup _{\left\Vert x \right\Vert = 1}}\{ {\left\Vert {Tx} \right\Vert^2} - \vert(Tx,x){\vert^2}\} \geqslant R_T^2$, where $ {R_T}$ is the radius of the smallest disc containing the spectrum of T. An example is given, which shows that we cannot expect equality for $ {G_1}$ operators in general, contrary to a conjecture of Istraţescu.

References [Enhancements On Off] (What's this?)

  • [1] G. Björck and V. Thomée, A property of bounded normal operators in Hilbert space, Ark. Math. 4 (1963), 551-555. MR 0149308 (26:6798)
  • [2] S. Hildebrandt, Über den numerischen Wertebereich eines Operators, Math. Ann. 164 (1966), 230-247. MR 0200725 (34:613)
  • [3] V. Istraţescu, On a class of operators, Math. Z. 124 (1972), 199-203 MR 0291854 (45:944)
  • [4] I. H. Sheth, On a conjecture of Istraţescu, J. Indian Math. Soc. 38 (1974), 337-338. MR 0402519 (53:6338)

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Keywords: Spectrum, $ {G_1}$ operator
Article copyright: © Copyright 1980 American Mathematical Society

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