View source for Positive-definite operator

A [[Symmetric operator|symmetric operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p0739101.png" /> on a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p0739102.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p0739103.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p0739104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p0739105.png" />. Any positive-definite operator is a [[Positive operator|positive operator]].



====Comments====
More generally, a positive-definite operator is defined as a bounded symmetric (i.e. self-adjoint) operator such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p0739106.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p0739107.png" />. This includes the diagonal operator, which acts on a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p0739108.png" /> of a Hilbert space as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p0739109.png" />. A non-negative-definite operator is one for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p07391010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073910/p07391011.png" />, cf. [[#References|[a2]]]. Sometimes a non-negative-definite operator is called a [[Positive operator|positive operator]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hille,   "Methods in classical and functional analysis" , Addison-Wesley  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators" , '''1–3''' , Interscience  (1958–1971)  pp. 906</TD></TR></table>