Various form closures associated with a fixed non-semibounded self-adjoint operator

Abstract

If $T$ is a semibounded self-adjoint operator in a Hilbert space $(H, \, (\cdot , \cdot))$ then the closure of the sesquilinear form $(T \cdot , \cdot)$ is a unique Hilbert space completion. In the non-semibounded case a closure is a Kre\u{\i}n space completion and generally, it is not unique. Here, all such closures are studied. A one-to-one correspondence between all closed symmetric forms (with ``gap point'' $0$) and all J-non-negative, J-self-adjoint and boundedly invertible Kre\u{\i}n space operators is observed. Their eigenspectral functions are investigated, in particular near the critical point infinity. An example for infinitely many closures of a fixed form $(T \cdot , \cdot)$ is discussed in detail using a non-semibounded self-adjoint multiplication operator $T$ in a model Hilbert space. These observations indicate that closed symmetric forms may carry more information than self-adjoint Hilbert space operators.