Representation of non-semibounded quadratic forms and orthogonal additivity Articles uri icon

publication date

  • March 2021

start page

  • 1

end page

  • 25


  • 2


  • 495

International Standard Serial Number (ISSN)

  • 0022-247X

Electronic International Standard Serial Number (EISSN)

  • 1096-0813


  • 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.


  • Mathematics


  • representations non-semibounded quadratic forms; direct integrals; orthogonal additivity; spectral theorem