Diagonalization of operators over continuous fields of $C^*$-algebras

A proof is given of a non-commutative analogue of the classical Hilbert–Schmidt theorem on diagonalization of a self-adjoint compact operator in a Hilbert space; namely, it is shown for a certain class of $C^*$-algebras that a self-adjoint compact operator in a Hilbert module $H_A$ over a $C^*$-algebra $A$ can be reduced to diagonal form in some larger module over a larger $W^*$-algebra, where the elements on the diagonal belong to $A$.