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Pseudo Numerical Ranges and Spectral Enclosures.
Borbala Gerhat1, Christiane Tretter1
1Math. Institut, Universität Bern, Sidlerstr. 5, 3012 Bern, Switzerland.
Complex Analysis and Operator Theory
|July 5, 2022
Summary
This study introduces pseudo numerical ranges and pseudo block numerical ranges for operator functions and matrices. These new concepts provide improved spectral enclosures for operator matrices and damped wave equations.
Area of Science:
- Operator theory
- Functional analysis
- Spectral theory
Background:
- Existing numerical ranges are limited, especially for unbounded operators.
- Operator polynomials and families of sesquilinear forms require advanced analysis.
Purpose of the Study:
- Introduce pseudo numerical range and pseudo block numerical range for operator functions and matrices.
- Extend these concepts to unbounded operators and operator families.
- Investigate spectral inclusion properties and applications.
Main Methods:
- Definition and analysis of pseudo numerical range for operator functions.
- Definition and analysis of pseudo block numerical range for operator matrices.
- Exploration of spectral inclusion properties for these new ranges.
Main Results:
- Established spectral inclusion properties for pseudo numerical and pseudo block numerical ranges.
- Developed spectral enclosures for operator matrices using Schur complements, relaxing dominance conditions.
- Derived new spectral bounds for linearly damped wave equations.
Conclusions:
- Pseudo numerical ranges offer powerful tools for analyzing operator functions and matrices.
- The new spectral enclosures provide broader applicability to operator matrices.
- The findings advance the understanding of spectral properties in damped wave equations.
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