Numerical range and compressions of the shift
| Autor: | Kelly Bickel, Pamela Gorkin |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Mathematics - Complex Variables Hilbert space Hardy space Shift operator Functional Analysis (math.FA) Mathematics - Functional Analysis Linear map symbols.namesake Operator (computer programming) Bounded function Primary 47A12 Secondary 47A13 30C15 FOS: Mathematics symbols Complex Variables (math.CV) Connection (algebraic framework) Numerical range Mathematics |
| Zdroj: | Complex Analysis and Spectral Theory. :241-261 |
| ISSN: | 1098-3627 0271-4132 |
| DOI: | 10.1090/conm/743/14964 |
| Popis: | The numerical range of a bounded, linear operator on a Hilbert space is a set in $\mathbb{C}$ that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space $H^2(\mathbb{D})$, and compressions of the shift operator to model spaces, i.e.~spaces of the form $H^2 \ominus \theta H^2$ where $\theta$ is inner. For these compressions of the shift operator, we provide a survey of results on the connection between their numerical ranges and the numerical ranges of their unitary dilations. We also discuss related results for compressed shift operators on the bidisk associated to rational inner functions and conclude the paper with a brief discussion of the Crouzeix conjecture. Comment: Survey paper, 21 pages |
| Databáze: | OpenAIRE |
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