We deal with homogeneous Besov and Triebel-Lizorkin spaces in the setting of doubling metric measure space in the presence of a non-negative self-adjoint whose heat kernel has Gaussian localization and the Markov property. class of almost diagonal operators on the associated sequence spaces is and it is shown that this class is an algebra. The boundedness of diagonal operators is utilized for establishing smooth molecular and decompositions for the above homogeneous Besov and Triebel-Lizorkin. Spectral multipliers for these spaces are established as well.