Related to spectrum

Question

Let $T$ be a bounded self-adjoint operator on $\mathcal{H}$, let $\mathcal{K}$ be a closed invariant subspace under $T$. Can we say $\sigma(T|\mathcal{K})\subset \sigma(T)$? Where $\sigma$ stands for spectrum?

Answer 1

Notice that regardless of any spectrum assumption, $T$ maps vectors orthogonal to $\mathcal{K}$ (let us denote $K’$ this state) to elements of $K’$. Let $P$ be the orthogonal projection to $\mathcal{K}$. This implies that $T-\lambda I$ commutes with $P$ for any scalar $\lambda$. Then, should $T-\lambda I$ be invertible, its inverse would also commute with $P$, and thus map $\mathcal{K}$ into itself. Thus we would have $(T-\lambda I)_{|\mathcal{K}}$ invertible.