Representation of non-semibounded quadratic forms and orthogonal additivity

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:495

Article

Ibort, Alberto^1,2  Llavona, Jose G.³  Lledo, Fernando^1,2  Manuel Perez-Pardo, Juan^1,2 

[1] Univ Carlos III Madrid, Avda Univ 30, Leganes 28911, Madrid, Spain

[2] Inst Ciencias Matemat CSIC UAM UC3M UCM, Nicolas Cabrera 13-15, Canto Blanco 28049, Madrid, Spain

[3] Univ Complutense Madrid, Plaza Ciencias 3, Madrid 28040, Spain

关键词: Representations non-semibounded quadratic forms;    Direct integrals;    Orthogonal additivity;    Spectral theorem;   

DOI  :  10.1016/j.jmaa.2020.124783

来源: Elsevier

【 摘 要 】

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the position operator in quantum mechanics and quadratic forms invariant under a unitary representation of a separable locally compact group. The case of invariance under a compact group is also discussed in detail. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free