On joint numerical radius
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- Proc. Amer. Math. Soc. 142 (2014), 1371-1380 Request permission
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 $|\langle T_jx,x\rangle |$ 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 $w(T)\ge \|T\|/2$.References
- 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
- Thøger Bang, A solution of the “plank problem.”, Proc. Amer. Math. Soc. 2 (1951), 990–993. MR 46672, DOI 10.1090/S0002-9939-1951-0046672-4
- 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, DOI 10.1016/S1874-5849(01)80006-1
- 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-New York, 1971. MR 0288583
- 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, DOI 10.1080/03081088708817806
- 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, DOI 10.1016/j.laa.2003.06.011
- 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, DOI 10.4064/sm194-1-6
- 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
Additional Information
- Vladimir Müller
- Affiliation: Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Praha 1, Czech Republic
- Email: muller@math.cas.cz
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1371-1380
- MSC (2010): Primary 47A12; Secondary 47A13
- DOI: https://doi.org/10.1090/S0002-9939-2014-11876-4
- MathSciNet review: 3162257