# A Hilbert Space Operator Representation of Abelian Po-Groups of Bilinear Forms

Research paper by Jiří Janda, Jan Paseka

Indexed on: 05 Feb '15Published on: 05 Feb '15Published in: International Journal of Theoretical Physics

#### Abstract

The existence of a non-trivial singular positive bilinear form Simon (J. Funct. Analysis 28, 377–385 (1978)) yields that on an infinite-dimensional complex Hilbert space $${\mathcal {H}}$$ the set of bilinear forms $${\mathcal {F}}(\mathcal {H})$$ is richer than the set of linear operators $${\mathcal {V}}(\mathcal {H})$$. We show that there exists an structure preserving embedding of partially ordered groups from the abelian po-group $${\mathcal {S}}_{D}(\mathcal {H})$$ of symmetric bilinear forms with a fixed domain D on a Hilbert space $${\mathcal {H}}$$ into the po-group of linear symmetric operators on a dense linear subspace of an infinite dimensional complex Hilbert spacel2(M). Moreover, if we restrict ourselves to the positive parts of the above mentioned po-groups, we can embed positive bilinear forms into corresponding positive linear operators.