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evilpostingmong
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positive operator proof
Prove that if T ∈ L(V) is positive, then so is Tk for every positive
integer k.
Let v=b1v1+...+bnvn. Now since T is positive, T has a positive square root. T=S^2. <S^2v, v>=<S^2v1, v>+...+<S^2vn, v>. Now <S^4v, v>=<S^2v1, v>^2+...+<S^2vn, v>^2 yes? Now since S^2 is positive, S^4 is positive. And since S^4 is>=S^2,
<S^4v1, v>+...+<S^4vn, v> is >=<S^2v1, v>+...+<S^2vn, v> which is >=0.
But this doesn't show that its true for S^2k. I remember learning about a technique
called induction. Should that be used?
Homework Statement
Prove that if T ∈ L(V) is positive, then so is Tk for every positive
integer k.
Homework Equations
The Attempt at a Solution
Let v=b1v1+...+bnvn. Now since T is positive, T has a positive square root. T=S^2. <S^2v, v>=<S^2v1, v>+...+<S^2vn, v>. Now <S^4v, v>=<S^2v1, v>^2+...+<S^2vn, v>^2 yes? Now since S^2 is positive, S^4 is positive. And since S^4 is>=S^2,
<S^4v1, v>+...+<S^4vn, v> is >=<S^2v1, v>+...+<S^2vn, v> which is >=0.
But this doesn't show that its true for S^2k. I remember learning about a technique
called induction. Should that be used?
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