Invariant Under Adjoint

Definition

• Let $V$ be a finite dimensional Inner Product Space
• Let $T\in \mathcal{L}(V)$
• Suppose $W$ is a subspace of $V$
• Then, $W^{\perp }$ is Invariant under $T^{\ast }$

Proof

1. Let $\alpha \in W$
2. Let $\beta \in W^{\perp }$
3. Since $\alpha \in W,T\alpha \in W⟹⟨T\alpha |\beta ⟩=0$
4. Thus, $⟨\alpha |T^{\ast }\beta ⟩=0$
5. Since $⟨T\alpha |\beta ⟩=⟨\alpha |T\beta ⟩$
6. Thus, if $\beta \in W^{\perp }$, Then, $T^{\ast }\beta \in W^{\perp }$