Some aspects of the invariant subspace problem (Q2760179)

Suppose that \(T\) is a linear operator on a vector space \(X\). A subspace \(M\) of \(X\) is called an invariant subspace of \(T\) if \(T(M)\subset M\). An invariant subspace \(M\) of \(T\) is nontrivial if \(M\) is different from \(\{0\}\) and \(M\) is different from \(X\). The invariant subspace problem asks whether or not every operator has a nontrivial invariant subspace, under various assumptions on \(X\). For example, if \(1< \dim(X)<\infty\), then every operator \(T\) on \(X\) has a nontrivial invariant subspace; just consider eigenvalues and eigenspaces. It is by now well-known that there exist bounded linear operators on Banach spaces that do not possess nontrivial closed invariant subspaces. However, the invariant subspace problem is still open in the Hilbert space case.NEWLINENEWLINENEWLINEThe paper under review is a survey of several ideas and results on the theory of invariant subspaces, both in the Banach and in the Hilbert space case. The authors are major players in this field and the paper also contains a historical account of various constructions of counterexamples to the invariant subspace problem in the Banach space setting.NEWLINENEWLINEFor the entire collection see [Zbl 0970.46001].