The classical Sturm comparison and separation theorems have been generalized by M. Picone and by G. Cimmino to partial differential equations and to systems of ordinary and partial differential equations [1]. In [2] G. Cimmino has proved the Sturm theorems for mappings valued in $l^{2}$. Recently several authors [3, 4, 5, 6, 7] have studied the Sturm theorems for ordinary and partial equations with solutions taking their values in $\mathbf{R}$ or $\mathbf{R}^{n}$. C. Pontini [8] has proved a Sturm theorem for ordinary equations whose solutions take their values in a Banach algebra. In my work I prove Sturm comparison and separation theorems for linear homogenous selfadjoint partial differential equations of the second order whose solutions take their values in a Hilbert space. I suppose that a non-trivial solution of a differential equation taking its values in a Hilbert space vanishes on the boundary of a smooth domain $D \subset \mathbf{R}^{n}$ and I prove that a solution of another related differential equation with solutions taking their values in $\mathcal{L}(H)$ has not an inverse in at least a point of $D$. Fundamental theorems are (2) and (3) from which we obtain the identity (4) which is the basis for Sturm theorems.