# Homework Help: Adjoint of linear Operator and T-invariant subspace

1. Oct 30, 2011

### H12504106

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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.