LAUSR

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