Frechet–Hilbert spaces and the property SCBS

Abstract

In this note, we obtain that all separable Frechet–Hilbert spaces have the property of smallness up to acomplemented Banach subspace (SCBS). Djakov, Terzioğlu, Yurdakul, and Zahariuta proved that a bounded perturbationof an automorphism on Fr´echet spaces with the SCBS property is stable up to a complemented Banach subspace.Considering Frechet–Hilbert spaces we show that the bounded perturbation of an automorphism on a separable Frechet–Hilbert space still takes place up to a complemented Hilbert subspace. Moreover, the strong dual of a real Frechet–Hilbertspace has the SCBS property.