(adsbygoogle = window.adsbygoogle || []).push({}); 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[itex]_{W}[/itex])* = (T*)[itex]_{W}[/itex]. Note that T[itex]_{W}[/itex] denotes the restriction of T to W

2. Relevant equations

[itex]\forall[/itex]x[itex]\in[/itex]W, T[itex]_{W}[/itex](x) = T(x)

T(W)[itex]\subseteq[/itex]W

T*(W)[itex]\subseteq[/itex]W

3. The attempt at a solution

I attempt to show that the image of any x [itex]\in[/itex]W under (T[itex]_{W}[/itex])* and (T*)[itex]_{W}[/itex] are the same.

<T[itex]_{W}[/itex]*(x), y> = < x, T[itex]_{W}[/itex](y)> = < x, T(y) > = < T*(x), y> = <(T*)[itex]_{W}[/itex](x), y >

Since this is true for all y, therefore, (T[itex]_{W}[/itex])* = (T*)[itex]_{W}[/itex].

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

Thank You.

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

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