Positive-definite operator

A symmetric operator $A$ on a Hilbert space $H$ such that

$$\inf \frac{\langle Ax, x \rangle }{\langle x, x \rangle } > 0$$

for any $x \in H$, $x \neq 0$. Any positive-definite operator is a positive operator.

Comments

More generally, a positive-definite operator is defined as a bounded symmetric (i.e. self-adjoint) operator such that $\langle Ax, x\rangle > 0$ for all $x \neq 0$. This includes the diagonal operator, which acts on a basis $( e _ {n} ) _ {n=} 1 ^ \infty$ of a Hilbert space as $Ae _ {n} = n ^ {-} 1 e _ {n}$. A non-negative-definite operator is one for which $\langle Ax, x \rangle \geq 0$ for all $x \in H$, cf. [a2]. Sometimes a non-negative-definite operator is called a positive operator.

References

[a1]	E. Hille, "Methods in classical and functional analysis" , Addison-Wesley (1972)
[a2]	N. Dunford, J.T. Schwartz, "Linear operators" , 1–3 , Interscience (1958–1971) pp. 906