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 Kreĭn 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 Kreĭn 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.
