Non-standard behaviour of semibounded forms and their representations

Abstract

A number of results is presented concerning the extreme behaviour of products of linear operators or relations from one Hilbert space H to another Hilbert space K. It is shown that there exists a relation T from H to K such that the closure of the nonnegative relation T^*T in H is not equal to the nonnegative selfadjoint relation T^*T^**; moreover, this may happen for a singular but also for a regular (i.e., closable) relation T. In fact, as an extreme situation the paper contains a construction of a densely defined closable operator T, such that dom T^*T={0}, while T^*T^** is a densely defined selfadjoint operator. Likewise it is shown that there exist a linear relation T from H to K and a nonnegative everywhere defined contraction R in K for which the product RT is singular and the product R^1/2 T is regular. Such results illustrate situations that can arise in the Lebesgue type decomposition of linear relations and the Lebesgue type decomposition of semibounded forms.

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BibTeX RIS

BibTeX

@misc{hassi2026nonstandard,
  title = {{Non-standard behaviour of semibounded forms and their representations}},
  author = {Seppo Hassi and Henk de Snoo},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08225},
  eprint = {2611.08225},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08225},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Non-standard behaviour of semibounded forms and their representations
AU  - Seppo Hassi
AU  - Henk de Snoo
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08225
UR  - https://arxiv.org/abs/2611.08225
AB  - A number of results is presented concerning the extreme behaviour of products of linear operators or relations from one Hilbert space H to another Hilbert space K. It is shown that there exists a relation T from H to K such that the closure of the nonnegative relation T^*T in H is not equal to the nonnegative selfadjoint relation T^*T^**; moreover, this may happen for a singular but also for a regular (i.e., closable) relation T. In fact, as an extreme situation the paper contains a construction of a densely defined closable operator T, such that dom T^*T={0}, while T^*T^** is a densely defined selfadjoint operator. Likewise it is shown that there exist a linear relation T from H to K and a nonnegative everywhere defined contraction R in K for which the product RT is singular and the product R^1/2 T is regular. Such results illustrate situations that can arise in the Lebesgue type decomposition of linear relations and the Lebesgue type decomposition of semibounded forms.
ER  -

Abstract

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