Restrictions of relatively Tauberian operators. (Q1417982)

The author considers a spectral condition under which a bounded linear operator \(T\) on a Banach space \(X\) with the property \(\overline{TX}_1\subset TX\) satisfies the condition \(\overline{TY}_1\subset TY\), where \(Y\) is a closed \(T\)-invariant subspace, also \(X_1\) and \(Y_1\) are respectively the closed unit balls of \(X\) and \(Y\).