Journal article
Completing an Operator Matrix and the Free Joint Numerical Radius
Complex analysis and operator theory, v 16(8), 114
01 Nov 2022
Abstract
Ando's classical characterization of the unit ball in the numerical radius norm was generalized by Farenick, Kavruk, and Paulsen using the free joint numerical radius of a tuple of Hilbert space operators (X-1, .. , X-m). In particular, the characterization leads to a positive definite completion problem. In this paper, we study various aspects of Ando's result in this generalized setting. Among other things, this leads to the study of finding a positive definite solution L to the equation L = I + Sigma(m )(j=1)[((LXj)-X-1/2*LXjL(1/2) + 1/4 I)(1/2 )+ ( L-1/2 XjLXj*L-1/2 + 1/4I)(1/2)], which may be viewed as a fixed point equation. Once such a fixed point is identified, the desired positive definite completion is easily obtained. Along the way we derive other related results including basic properties of the free joint numerical radius and an easy way to determine the free joint numerical radius of a tuple of generalized permutations. Finally, we present some open problems.
Metrics
Details
- Title
- Completing an Operator Matrix and the Free Joint Numerical Radius
- Creators
- Kennett L. Dela Rosa - Univ Philippines Diliman, Inst Math, Quezon City 1101, Ncr, PhilippinesHugo J. Woerdeman - Drexel University, Mathematics
- Publication Details
- Complex analysis and operator theory, v 16(8), 114
- Publisher
- Springer Nature
- Number of pages
- 34
- Grant note
- DMS 2000037 / National Science Foundation; National Science Foundation (NSF) Office of the Chancellor of the University of the Philippines Diliman, through the Office of the Vice Chancellor for Research and Development 355645 / Simons Foundation
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000878676500001
- Scopus ID
- 2-s2.0-85141169980
- Other Identifier
- 991021861176604721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics
- Mathematics, Applied