SPECTRALITY FOR MATRICES OF FOURIER MULTIPLIER OPERATORS ACTING IN L^p -SPACES OVER LCA GROUPS

Abstract

N. Dunford and J.T. Schwartz gave a complete characterization of those matrices of (bounded) Fourier multiplier operators acting in L ²(R^N) ⁿ which are spectral operators, [4; ch. XV]. In the present note this characterization is extended to the setting of L^P(G)ⁿ , where G is a locally compact abelian group and 1 < p < ∞; see Theorem 2.