(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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Adjoint of linear Operator and T-invariant subspace

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**