Perturbation theory of linear operators

Remarks

Example: every bounded operator $V$ on $X$ is $T$-bounded with $T$-bound $0$, since $\norm{Vx} \le \norm{V}\norm{x}$. Non-example: on $L^2(\R)$, let $T = -i \, d/dx$ with domain the Sobolev space $H^1(\R)$, and let $V = -d^2/dx^2$ with domain $H^2(\R)$. Clause 1 fails, since $H^1(\R)$ is not contained in $H^2(\R)$.

A basic result is the Kato–Rellich theorem. If $T$ is self-adjoint on a Hilbert space and $V$ is symmetric and $T$-bounded with $T$-bound less than $1$, then $T + V$ is self-adjoint on $D(T)$. Another is Weyl's theorem: a compact perturbation does not change the essential spectrum of a self-adjoint operator.