Non-semibounded closed symmetric forms associated with a generalized Friedrichs extension

The theory of closed sesquilinear forms in the non-semibounded situation exhibits some new features, as opposed to the semibounded situation. In particular, there can be more than one closed form associated with the generalized Friedrichs extension SF of a non-semibounded symmetric operator S (if SF exists). However, there is one unique form tF [•; •] satisfying Kato's second representation theorem and, in particular, dom tF = dom jSF j1=2. In the present paper another closed form tF [•; •] is constructed which is also uniquely associated with SF . The relation between these two forms is analyzed and it is shown that these two non-semibounded forms can indeed differ from each other. Some general criteria for their equality are established. The results induce solutions to some open problems concerning generalized Friedrichs extensions and complete some earlier results about them in the literature. The study is connected to the spectral functions of definitizable operators in Krein spaces.


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