Complete spectral sets and numerical range
| Autor: | Kenneth R. Davidson, Hugo J. Woerdeman, Vern I. Paulsen |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
47A12
47A25 15A60 Discrete mathematics Physical constant Applied Mathematics General Mathematics Operator (physics) 010102 general mathematics Mathematics - Operator Algebras 010103 numerical & computational mathematics 01 natural sciences Matrix polynomial Operator algebra Norm (mathematics) Bounded function FOS: Mathematics Homomorphism 0101 mathematics Operator Algebras (math.OA) Numerical range Mathematics |
| Zdroj: | Proceedings of the American Mathematical Society. 146:1189-1195 |
| ISSN: | 1088-6826 0002-9939 |
| Popis: | We define the complete numerical radius norm for homomorphisms from any operator algebra into B ( H ) \mathcal B(\mathcal H) , and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if K K is a complete C C -spectral set for an operator T T , then it is a complete M M -numerical radius set, where M = 1 2 ( C + C − 1 ) M=\frac 12(C+C^{-1}) . In particular, in view of Crouzeix’s theorem, there is a universal constant M M (less than 5.6) so that if P P is a matrix polynomial and T ∈ B ( H ) T \in \mathcal B(\mathcal H) , then w ( P ( T ) ) ≤ M ‖ P ‖ W ( T ) w(P(T)) \le M \|P\|_{W(T)} . When W ( T ) = D ¯ W(T) = \overline {\mathbb D} , we have M = 5 4 M = \frac 54 . |
| Databáze: | OpenAIRE |
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