An application of the smooth variational principle to the existence of nontrivial invariant subspaces
The main result of the paper under review is the following theorem: Let \(X\) be a real Banach space which has an equivalent Gâteaux smooth norm and let \(F\) be a nontrivial closed subset of \(X\). Let \(\mathcal S\) be a semigroup of (bounded linear) operators on \(X\). If for all \(A\in\mathcal S\) there is a differentiable function \(f_A\:\mathcal J_A=(-\delta_A,\delta_A)\to L(X)\) such that (i)~\(f_A(0)=I\), (ii)~\((f_A)'(0)=A\), (iii)~\(F\) is an invariant subset of each element of \(f_A(\mathcal I_A)\), then there is a nontrivial invariant subspace for \(\mathcal S\). NEWLINENEWLINENEWLINEUsing this theorem, some invariant subspace results are proved in the paper, e.g., an operator \(X\) which admits a moment sequence has a nontrivial invariant subspace.
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