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evilpostingmong
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Homework Statement
Suppose that T is a positive operator on V. Prove that T is invertible
if and only if <Tv,v > is >0 for every v ∈ V \ {0}.
Homework Equations
The Attempt at a Solution
If T is invertible, then TT-1=I.Now let v=v1+...+vn and let Tv=a1v1+...+anvn. Now <Tv, v>=<a1v1, v>+...+<anvn, v>. Applying T-1 we get <T-1(a1v1), v>+...+<T-1(anvn),v> =<v1, v>+...+<vn, v>=<Iv, v>=<v, v>. Since v[tex]\notin[/tex]{0},<v, v> is >0. And since T is invertible,
<Tv, v>=<v, v> if T=I, or <Tv, v> > <v, v> if T=/=I. Therefore <Tv, v> is >=<v, v>.
And since <v, v> is >0, <Tv, v> is >0.