Shape, scale, and minimality of matrix ranges
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Abstract:
We study containment and uniqueness problems concerning matrix convex sets. First, to what extent is a matrix convex set determined by its first level? Our results in this direction quantify the disparity between two product operations, namely, the product of the smallest matrix convex sets over $K_i \subseteq \mathbb {C}^d$ and the smallest matrix convex set over the product of $K_i$. Second, if a matrix convex set is given as the matrix range of an operator tuple $T$, when is $T$ determined uniquely? We provide counterexamples to results in the literature, showing that a compact tuple meeting a minimality condition need not be determined uniquely, even if its matrix range is a particularly friendly set. Finally, our results may be used to improve dilation scales, such as the norm bound on the dilation of (not necessarily self-adjoint) contractions to commuting normal operators, both concretely and abstractly.References
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Additional Information
- Benjamin Passer
- Affiliation: Faculty of Mathematics, Technion-Israel Institute of Technology, 3200003 Haifa, Israel
- Address at time of publication: Department of Pure Mathematics, University of Waterloo, Ontario, Canada
- MR Author ID: 1083708
- Email: benjaminpas@technion.ac.il; bpasser@uwaterloo.ca
- Received by editor(s): March 29, 2018
- Received by editor(s) in revised form: June 19, 2018
- Published electronically: April 25, 2019
- Additional Notes: The author was partially supported by a Zuckerman Fellowship at the Technion.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 1451-1484
- MSC (2010): Primary 47A20, 47A13, 46L07, 47L25
- DOI: https://doi.org/10.1090/tran/7665
- MathSciNet review: 3968808