The concept of a Hilbert network, introduced in another paper, extends network theory to finite or infinite networks whose elements are described by operators on a Hilbert space. The present work investigates a variety of qualitative properties possessed by such networks. In particular, some operators associated with the entire network, such as the driving-point impedances as well as the admittance operator which relates the branch-voltage vector to the branch-current vector, are shown to be—under suitable conditions—either positive, monotonic, or convex. Also, generalized versions of Jeans’ least power theorem and the Shannon–Hagelbarger theorem are proved. Finally, a bound on the power dissipation is determined.