Coercive Function - Coercive Operators and Forms

Coercive Operators and Forms

A self-adjoint operator where is a real Hilbert space, is called coercive if there exists a constant such that

for all in

A bilinear form is called coercive if there exists a constant such that

for all in

It follows from the Riesz representation theorem that any symmetric ( for all in ), continuous ( for all in and some constant ) and coercive bilinear form has the representation

for some self-adjoint operator which then turns out to be a coercive operator. Also, given a coercive operator self-adjoint operator the bilinear form defined as above is coercive.

One can also show that any self-adjoint operator is a coercive operator if and only if it is a coercive function (if one replaces the dot product with the more general inner product in the definition of coercivity of a function). The definitions of coercivity for functions, operators, and bilinear forms are closely related and compatible.