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On joint numerical radius

Author: Vladimir Müller
Journal: Proc. Amer. Math. Soc. 142 (2014), 1371-1380
MSC (2010): Primary 47A12; Secondary 47A13
Published electronically: January 29, 2014
MathSciNet review: 3162257
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Abstract: Let $ T_1,\dots ,T_n$ be bounded linear operators on a complex Hilbert space $ H$. We study the question whether it is possible to find a unit vector $ x\in H$ such that $ \vert\langle T_jx,x\rangle \vert$ is large for all $ j$. Thus we are looking for a generalization of a well-known fact for $ n=1$ that the numerical radius $ w(T)$ of a single operator $ T$ satisfies $ w(T)\ge \Vert T\Vert/2$.

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  • [AT] Yik Hoi Au-Yeung and Yiu Tung Poon, A remark on the convexity and positive definiteness concerning Hermitian matrices, Southeast Asian Bull. Math. 3 (1979), no. 2, 85-92. MR 564798 (81c:15026)
  • [B] Thøger Bang, A solution of the ``plank problem'', Proc. Amer. Math. Soc. 2 (1951), 990-993. MR 0046672 (13,769a)
  • [Ba] Keith Ball, Convex geometry and functional analysis, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 161-194. MR 1863692 (2003c:52001),
  • [BD] F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Mathematical Society Lecture Note Series, vol. 2, Cambridge University Press, London, 1971. MR 0288583 (44 #5779)
  • [FT] M. K. H. Fan and A. L. Tits, On the generalized numerical range, Linear and Multilinear Algebra 21 (1987), no. 3, 313-320. MR 928286 (89b:16031),
  • [GJK] Eugene Gutkin, Edmond A. Jonckheere, and Michael Karow, Convexity of the joint numerical range: topological and differential geometric viewpoints, Linear Algebra Appl. 376 (2004), 143-171. MR 2014890 (2004i:15021),
  • [LP] Chi-Kwong Li and Yiu-Tung Poon, The joint essential numerical range of operators: convexity and related results, Studia Math. 194 (2009), no. 1, 91-104. MR 2520042 (2010c:47012),
  • [M] Vladimir Müller, Spectral theory of linear operators and spectral systems in Banach algebras, 2nd ed., Operator Theory: Advances and Applications, vol. 139, Birkhäuser Verlag, Basel, 2007. MR 2355630 (2008g:47013)

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Additional Information

Vladimir Müller
Affiliation: Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Praha 1, Czech Republic

Keywords: Joint numerical range, numerical radius
Received by editor(s): November 16, 2011
Received by editor(s) in revised form: May 23, 2012
Published electronically: January 29, 2014
Additional Notes: This research was supported by grants 201/09/0473 of GA ČR, IAA100190903 of GA AV ČR and RVO: 67985840
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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