Elliptic systems of pseudodifferential equations in a refined scale on a closed manifold

We study a system of pseudodifferential equations that is elliptic in the sense of Petrovskii on a closed compact smooth manifold. We prove that the operator generated by the system is Fredholm one on a refined two-sided scale of the functional Hilbert spaces. Elements of this scale are the special isotropic spaces of H{o}rmander--Volevich--Paneah.