Journal article
Completing an Operator Matrix and the Free Joint Numerical Radius
COMPLEX ANALYSIS AND OPERATOR THEORY, v 16(8), 114
Nov 2022
Featured in Collection : UN Sustainable Development Goals @ Drexel
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.
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Details
- Title
- Completing an Operator Matrix and the Free Joint Numerical Radius
- Publication Details
- COMPLEX ANALYSIS AND OPERATOR THEORY, v 16(8), 114
- Publisher
- SPRINGER BASEL AG; BASEL
- Grant note
- The first author acknowledges the Office of the Chancellor of the University of the Philippines Diliman, through the Office of the Vice Chancellor for Research and Development, for funding support through the Ph.D. Incentive Award. The research of the second author was supported by Simons Foundation grant 355645 and National Science Foundation grant DMS 2000037.
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Drexel University
- 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