Adjoint of linear Operator and T-invariant subspace

[H12504106]

Homework Statement

Let T be a linear operator on an inner product space V and W be a T-invariant subspace of V. If W is both T and T* invariant, prove that (T$_{W}$)* = (T*)$_{W}$. Note that T$_{W}$ denotes the restriction of T to W

Homework Equations

$\forall$x$\in$W, T$_{W}$(x) = T(x)
T(W)$\subseteq$W
T*(W)$\subseteq$W

The Attempt at a Solution

I attempt to show that the image of any x $\in$W under (T$_{W}$)* and (T*)$_{W}$ are the same.
<T$_{W}$*(x), y> = < x, T$_{W}$(y)> = < x, T(y) > = < T*(x), y> = <(T*)$_{W}$(x), y >

Since this is true for all y, therefore, (T$_{W}$)* = (T*)$_{W}$.

Is the above proof correct? I feel that there is something wrong in one of the steps.
Thank You.

 

[mighty2000]

Your proof looks correct to me. You have correctly shown that the images of any x in W under (T_W)* and (T*)_W are the same, which means that the two operators are equal. Well done!