An operator theoretic approach is used to solve the linearized Vlasov–Maxwell equations for transverse plasma oscillations. In particular, the special cases of simple and second‐order real zeros of the plasma dispersion function are treated and formulae for the amplitude of the plasma waves are presented. An existence and uniqueness theorem for the solution to the Vlasov–Maxwell transverse mode plasma equation is proved in an appendix. In a second appendix, a general characterization for the zeros of the plasma distribution function is presented for the case of any double humped equilibrium distribution.