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phagist_
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Homework Statement
Let [tex]T : V \rightarrow V[/tex] be a linear operator on a complex inner product space [tex]V[/tex] , and let
[tex]S = I + T^{*}T[/tex], where [tex]I : V \rightarrow V[/tex] is the identity.
(a) Write [tex]<Sx,x>[/tex] in terms of [tex]x[/tex] and [tex]Tx[/tex].
(b) Prove that every eigenvalue [tex]\lambda[/tex] of [tex]S[/tex] is real and satisfies [tex]\lambda[/tex][tex]\geq 1[/tex].
(c) Prove that the nullspace of [tex]S[/tex] is [tex]\{0\}[/tex].
Homework Equations
The Attempt at a Solution
Ok I really hardly have any idea on how to tackle this problem.
Am I right in thinking that; in terms of [tex]Tx[/tex] and [tex]x[/tex]
[tex]<Sx,x>[/tex] becomes [tex]<[I+T^{*}T]x,x>[/tex]?
If so, I'm thinking of using the properties of inner products, but so far I've only dealt with scalars inside the <,> brackets, and not linear operators.
I'm really struggling with it and thus haven't attempted parts b or c yet.
Any help/comments would be greatly appreciated,
cheers!
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