Corollary

Let $q$ be a closed positive partial form on E and $u$ the positive self-adjoint operator representing $q$. The following conditions are equivalent:

(i) The partial operator $u$ has compact resolvent;

(ii) The positive endomorphism $\surd\overline{(1_E + u)^{-1}}= (1_E+u)^{-1/2}$ of E is compact;

(iii) The canonical injection $j$ of the Hilbert space $E_q$ (Definition 9 of IV, p. 292) into E is compact.

Since $u$ is positive, the real number $-1$ belong to the resolvent set of $u$ (Proposition 17 of IV, p. 248), hence the positive endomorphism $v=\surd\overline{(1_E + u)^{-1}}$ is defined, and we have $v= (1_E+u)^{-1/2}$ by the functional calculus.

The endomorphism $v$ is compact if and only if $v^2= (1_E+u)^{-1}$ is compact (Proposition 6 of III, p. 91), that is to say if and only if $u$ has compact resolvent (Proposition 19, (iii)). This proves that conditions (i) and (ii) are equivalent.

Suppose that the endomorphism $v$ is compact. Let B be the unit ball of the Hilbert space $E_q$. Since $(1_E+u)^{1/2}$ is a continuous linear mapping from $E_q$ into E (Corollary of Theorem 3 of IV, p. 294), the subset $B'= (1_E+u)^{1/2}(B)$ of E is bounded, hence the subset $j(B) = B =v(B')$ is relatively compact in E (Remark 1 of III, p. 2). This proves that $j$ is compact. Thus (ii) implies (iii).

The linear mapping $\widetilde{v}:x\mapsto (1_E+u)^{-1/2}(x)$ from E into $E_q$ is well-defined and isometric (Corollary of Theorem 3 of IV, p. 294). Since $v=j\circ \widetilde{v}$, condition (iii) implies (ii) (Proposition 3 of III, p. 5).