Kato-Rellich theorem

Let $\mathscr{H}$ be a Hilbert space, $A:D(A)\subset \mathscr{H}\to \mathscr{H}$ a self-adjoint operator and $B:D(B)\subset \mathscr{H}\to \mathscr{H}$ a symmetric operator with $D(A)\subset D(B)$.

We say that $B$ is $A$-bounded if there are positive constants$\alpha ,\beta$ such that

$$\parallel Bx\parallel \le \alpha \parallel Ax\parallel +\beta \parallel x\parallel$$

for all$x\in D(A)$, and we say that $\alpha$ is an $A$-bound for $B$.

Theorem 1.

(Kato-Rellich) If $B$ is $A$-bounded with $A$-bound smallerthan $\mathrm{1}$, then $A\mathrm{+}B$ is self-adjoint on $D\mathit{}\mathrm{(}A\mathrm{)}$, and essentiallyself-adjoint on any core of $A$. Moreover, if $A$ is bounded below, thenso is $A\mathrm{+}B$.