On diagonal operators between the sequence (LF)-spaces $$l_p({\mathcal {V}})$$

Abstract Diagonal (multiplication) operators acting between a particular class of countable inductive spectra Fréchet sequence spaces, called (LF)-spaces, are investigated. We prove results concerning boundedness, compactness, power and mean ergodicity. Furthermore, we determine when diagonal operator is Montel reflexive. analyze the in terms system weights defining spaces.