Compressions of Linear Relations in Krein Spaces
Abstract
Given an orthogonal decomposition H̃ = H[dotplus]Ĥ of a Krein space, we study compressions CH(R) of linear relations R in H̃ to H. We formulate a sufficient condition that ensures that compression commutes with taking adjoints, that is, the adjoint identity CH(R*̃) = CH(R)* holds. We derive preservation and reflection results for the compression of maximal dissipative, maximal symmetric, and self-adjoint linear relations. For nonnegative linear relations, the adjoint identity also implies that compression commutes with taking the Friedrichs extension. For H≠{0}, we characterize finite dimensionality of the exit space Ĥ by universal compression properties. Finally, we give a criterion for the adjoint identity to hold in terms of bounded linear functionals on R.
Related
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BibTeX
@misc{curgus2026compressions,
title = {{Compressions of Linear Relations in Krein Spaces}},
author = {Branko Ćurgus and Aad Dijksma},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07299},
eprint = {2611.07299},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07299},
}
RIS
TY - UNPB
TI - Compressions of Linear Relations in Krein Spaces
AU - Branko Ćurgus
AU - Aad Dijksma
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07299
UR - https://arxiv.org/abs/2611.07299
AB - Given an orthogonal decomposition H̃ = H[dotplus]Ĥ of a Krein space, we study compressions C_H(R) of linear relations R in H̃ to H. We formulate a sufficient condition that ensures that compression commutes with taking adjoints, that is, the adjoint identity C_H(R^*̃) = C_H(R)^* holds. We derive preservation and reflection results for the compression of maximal dissipative, maximal symmetric, and self-adjoint linear relations. For nonnegative linear relations, the adjoint identity also implies that compression commutes with taking the Friedrichs extension. For H≠{0}, we characterize finite dimensionality of the exit space Ĥ by universal compression properties. Finally, we give a criterion for the adjoint identity to hold in terms of bounded linear functionals on R.
ER -