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1. Introduction. Many results from the theory of closed semi- bounded hermitian bilinear forms in a Hubert space can be extended so as to apply to forms which are not semibounded and indeed have no restriction placed on their numerical range. Some of these re­ sults are summarized in this note, while details can be found in (S). Our main interest here is in questions which arise naturally when the forms are assumed to be hermitian but not necessarily semibounded. Some such questions are: Is every closed hermitian form / of the type JA(U, V) = (A1/2U, Ain*v) with A selfadjoint? Does every sym­ metric operator B with equal deficiency indices have a selfadjoint extension A such that the form J A is minimal among those closed hermitian forms whose associated operators extend B? If such a minimal extension exists, is it necessarily unique? If our forms and operators are semibounded then each of these questions is known to have a positive answer (see (4, Chapter V)). We will show that in the general case the third question may have a negative answer, and that the second question can be answered in the affirmative when there is a gap in the essential spectrum of B. Complete answers to the first two questions are not known to the author. We remark that one reason for wanting to know whether a closed form is of the type J A is that results from interpolation and spectral theory can then be used to obtain information about the form from our knowledge of the operator A, 2. Closed forms. If X and F are complex linear spaces then a function / from X X F to the complex numbers is called a bilinear form (or more properly a sesquilinear form) if it is linear in the first variable and conjugate linear in the second. In the case when X and F are Hubert spaces and the bilinear form J is bounded (meaning | J(u, v)\ £y\\u\\x\\v\\Y for some positive y and all uÇzX and z;£ F), there exists a bounded linear operator C:X~*Y satisfying J(u> v) == (Cu, V)Y for uÇzX and vÇz F. If this operator C is an isomorphism

Hermitian bilinear forms which are not semibounded