A Note on Finite Rank Quadratic Operators

The representation result of a finite rank quadratic operator is given. It has been shown that the space FQ (H) of finite rank bounded quadratic operators on a real separable Hilbert space H is equal to the cross spaces L (H) H and SL (H) H for a space SL (H) of self-adjoint bonded linear operators on H and the space L (H) of bounded linear operators on H. As an application of the concept of finite rank quadratic operator, the numerical range of an operator ASL (H) is derived.