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

Title | Kato-Rellich theorem |
---|---|

Canonical name | KatoRellichTheorem |

Date of creation | 2013-03-22 14:52:59 |

Last modified on | 2013-03-22 14:52:59 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 7 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 47A55 |

Synonym | Rellich-Kato theorem |

Defines | A-bounded |

Defines | A-bound |