The vector space of tensor operators transforming according to the adjoint representation of the unitary group is investigated for arbitrary (even infinite) dimension of the group. In particular, a natural basis consisting of generalized number operators is constructed and some properties are studied. The results are applied to the construction of mass operators for arbitrary multiquark systems with arbitrary dimsension of the one-quark flavor space. It is shown, also, how the proved properties of the spectrum simplify the obtaining of certain mass formulas.