Self-adjoint operators on a Hilbert space define symmetric sesquilinear forms, but not every such form arises in this way, nor is the correspondence one-to-one in general, when one goes beyond the bounded case which is ruled by Riesz’s representation theorem.
The author extends the first representation theorem for indefinite forms to cover new cases (Theorem 2.3), building on the work of L. Grubišić et al. [Mathematika 59, No. 1, 169–189 (2013; Zbl 1272.47004)]. Spectral gap conditions on the representing operators are dropped, and new assumptions have to be imposed in view of Example 1. In the special case of non-negative forms with off-diagonal perturbations, a complete generalization of the results of [loc. cit.] is given (Theorem 2.7), together with an explicit description of the kernel (Theorem 2.13), extending the analysis of [V. Kostrykin et al., Oper. Theory Adv. Appl. 149, 349–372 (2004; Zbl 1084.47002)].
In order for the second representation theorem to hold as well, equivalent and sufficient conditions for the domain stability condition are presented (Theorem 3.8 and Lemma 3.9), as natural extensions of those contained in [Zbl 1272.47004)].