Extremal vectors and invariant subspaces

Abstract:

For a bounded linear operator on Hilbert space we define a sequence of so-called minimal vectors in connection with invariant subspaces and show that this presents a new approach to invariant subspaces. In particular, we show that for any compact operator $K$ some weak limit of the sequence of minimal vectors is noncyclic for all operators commuting with $K$ and that for any normal operator $N$, the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with $N$. Thus, we give a new and more constructive proof of existence of invariant subspaces. The sequence of minimal vectors does not seem to converge in norm for an arbitrary bounded linear operator. We will prove that if $T$ belongs to a certain class ${\mathcal {C}}$ of operators, then the sequence of such vectors converges in norm, and that if $T$ belongs to a subclass of ${\mathcal {C}}$, then the norm limit is cyclic.