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.
UYANIK, ELİF and YURDAKUL, MURAT HAYRETTİN
"Fréchet-Hilbert spaces and the property SCBS,"
Turkish Journal of Mathematics: Vol. 42:
3, Article 42.
Available at: https://journals.tubitak.gov.tr/math/vol42/iss3/42