as told on the topic i need to prove if U is a subspace of V and V is a vector space. $T\in L(V)$ which L is the set of all operators. I need to prove the both side of this statement below:
U is T invariant iff $U^\bot$ is T*-invariant
T* is adjoint operator of T and $U^\bot$ is orthogonal complement of U
i need this proof to use it to solve somthing else.
Suppose $TU\subseteq U$. Then for each $w\in U^\perp$, we have $\langle T^\ast w,u\rangle=\langle w,Tu\rangle=0$ for all $u\in U$. Hence $T^\ast w\in U^\perp$. Since $w$ is arbitrary, in turn we have $T^\ast U^\perp\subseteq U^\perp$. The argument for the other direction is similar.