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Normal Operator Proof
Prove an operator ##T \in L(V)## is normal ##⇔ ||T(v)|| = ||T^*(v)||##.
(1) ##T \in L(V)## is normal if ##TT^*= T^*T##.
(2) If T is a self-adjoint operator on V such that ##<T(v), v> = 0, \space \forall v \in V##, then ##T=0##.
##"\Rightarrow"## Assume ##T## is normal (1) :
##TT^*= T^*T##
##TT^* - T^*T = 0##
Now using (2) we can write :
##<(TT^* - T^*T)(v), v> = 0, \space \forall v \in V##
Using some inner product rules yields :
##<T^*T(v), v> = <TT^*(v), v>, \space \forall v \in V##
##||T(v)||^2 = ||T^*(v)||^2, \space \forall v \in V##
##||T(v)|| = ||T^*(v)||, \space \forall v \in V##
##"\Leftarrow"## : The proof will be exactly as above, except I start at ##||T(v)|| = ||T^*(v)||## and I finish at ##TT^*= T^*T## I believe?
Homework Statement
Prove an operator ##T \in L(V)## is normal ##⇔ ||T(v)|| = ||T^*(v)||##.
Homework Equations
(1) ##T \in L(V)## is normal if ##TT^*= T^*T##.
(2) If T is a self-adjoint operator on V such that ##<T(v), v> = 0, \space \forall v \in V##, then ##T=0##.
The Attempt at a Solution
##"\Rightarrow"## Assume ##T## is normal (1) :
##TT^*= T^*T##
##TT^* - T^*T = 0##
Now using (2) we can write :
##<(TT^* - T^*T)(v), v> = 0, \space \forall v \in V##
Using some inner product rules yields :
##<T^*T(v), v> = <TT^*(v), v>, \space \forall v \in V##
##||T(v)||^2 = ||T^*(v)||^2, \space \forall v \in V##
##||T(v)|| = ||T^*(v)||, \space \forall v \in V##
##"\Leftarrow"## : The proof will be exactly as above, except I start at ##||T(v)|| = ||T^*(v)||## and I finish at ##TT^*= T^*T## I believe?
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