Block numerical ranges for linear relations

@misc{aydi2026block,
  title = {{Block numerical ranges for linear relations}},
  author = {Rabeb Aydi and Hannes Gernandt and Omaima Kchaou and Carsten Trunk and Mitsuru Wilson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04431},
  eprint = {2610.04431},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04431},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Block numerical ranges for linear relations
AU  - Rabeb Aydi
AU  - Hannes Gernandt
AU  - Omaima Kchaou
AU  - Carsten Trunk
AU  - Mitsuru Wilson
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04431
UR  - https://arxiv.org/abs/2610.04431
AB  - 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.
ER  -