# 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 |

**How to Cite This Entry:**

Positive-definite operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_operator&oldid=48251