Representation Theorems for Indefinite Quadratic Forms Revisited
Luka Grubisic, Vadim Kostrykin, Konstantin A. Makarov, Kresimir, Veselic
TL;DR
This paper revisits fundamental representation theorems for indefinite quadratic forms, providing new proofs, conditions for their validity, and an explicit counterexample where the theorems fail.
Contribution
It offers simplified proofs of key theorems, establishes necessary and sufficient conditions for the second theorem, and presents a novel counterexample.
Findings
New straightforward proofs of representation theorems
Conditions for the second theorem's validity
Explicit example where the second theorem fails
Abstract
The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring the second representation theorem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.
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