Exercise 42, § 5

Let E be a complex Hilbert space and let $u$ be a closed partial operator with dense domain on E. A partial operator $v$ on E is said to be compact relative to $u$ if the domain of $v$ contains the domain of $u$ and if the restriction of $v$ to dom($u$) is a compact linear mapping of $E_u$ into E.

a) Every partial operator $v$ which is compact relative to $u$ is bounded relative to $u$ and its relative norm $\|v\|_u$ is zero. (If the relative norm is not zero, construct a sequence $(x_n)$ in dom($u$) converging to 0, such that $\|v(x_n)\|= 1$ for all $n$ and such that $(u(x_n))$ converges weakly to an element $y$ of E; prove that $y= 0$ and deduce that $v$ is not relatively compact.)

b) Let $\lambda \in \mathbf{C}-$ Sp($u$). Let $v$ be a partial operator on E whose domain contains dom($u$). Then $v$ is compact relative to $u$ if and only if $v\circ R(u, \lambda )$ is a compact endomorphism of E.