# Kato-Rellich theorem

Let $\mathcal{H}$ be a Hilbert space, $A\colon D(A)\subset\mathcal{H}\to\mathcal{H}$ a self-adjoint operator and $B\colon D(B)\subset\mathcal{H}\to\mathcal{H}$ 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

 $\|Bx\|\leq\alpha\|Ax\|+\beta\|x\|$

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 smaller than $1$, then $A+B$ is self-adjoint on $D(A)$, and essentially self-adjoint on any core of $A$. Moreover, if $A$ is bounded below, then so is $A+B$.

Title Kato-Rellich theorem KatoRellichTheorem 2013-03-22 14:52:59 2013-03-22 14:52:59 Koro (127) Koro (127) 7 Koro (127) Theorem msc 47A55 Rellich-Kato theorem A-bounded A-bound