Year: 2026
Type: preprint
Abstract: Linear relations are multivalued generalizations of linear operators whose spectral properties are closely linked to the stability of differential-algebraic equations. Consequently, reliable spectral enclosures are of considerable interest. In this note, we extend the established concept of the block numerical range from operators to linear relations by means of a range representation. Unlike the classical numerical range, the resulting block numerical range may be nonconvex and can therefore detect gaps in the spectrum. We establish fundamental properties of this new notion and derive corresponding spectral enclosure results. Finally, we illustrate the theory with a numerical example comparing the block numerical ranges obtained from different constructions. As an application, we obtain an explicit left-half-plane spectral bound for a class of port-Hamiltonian DAEs.
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W7220698368