An inequality concerning the smallest disc that contains the spectrum of an operator
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- by Gerhard Garske PDF
- Proc. Amer. Math. Soc. 78 (1980), 529-532 Request permission
Abstract:
Any bounded linear operator T on a Hilbert space satisfies the inequality ${\sup _{\left \| x \right \| = 1}}\{ {\left \| {Tx} \right \|^2} - |(Tx,x){|^2}\} \geqslant R_T^2$, 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 Istrǎţescu.References
- Göran Björck and Vidar Thomée, A property of bounded normal operators in Hilbert space, Ark. Mat. 4 (1963), 551–555. MR 149308, DOI 10.1007/BF02591603
- Stefan Hildebrandt, Über den numerischen Wertebereich eines Operators, Math. Ann. 163 (1966), 230–247 (German). MR 200725, DOI 10.1007/BF02052287
- Vasile I. Istrăţescu, On a class of normaloid operators, Math. Z. 124 (1972), 199–202. MR 291854, DOI 10.1007/BF01110797
- I. H. Sheth, On a conjecture of Istrătescu, J. Indian Math. Soc. (N.S.) 38 (1974), no. 1, 2, 3, 4, 337–338 (1975). MR 402519
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 78 (1980), 529-532
- MSC: Primary 47A10
- DOI: https://doi.org/10.1090/S0002-9939-1980-0556626-5
- MathSciNet review: 556626