Backward minimal points for bounded linear operators on finite-dimensional vector spaces

Abstract

For a bounded linear operator T:H→H with dense range on the Hilbert space H,x 0∈H and ∥x 0∥>ϵ>0 , the backward minimal point y(n) is the unique vector of smallest norm in the set {y:∥T ny−x 0∥⩽ϵ} . We investigate the limit of the sequence (T ny(n)) for operators T on finite-dimensional vector spaces. This vector—lim T ny(n) —is used by Ansari and Enflo in [Trans. Amer. Math. Soc. 350 (1998) 539] to construct invariant subspaces for compact and normal operators in infinite dimensions. Here, we find a geometric description of this vector for invertible normal operators on C n and self-adjoint operators on R n with orthogonal eigenvectors. We also show that the sequence (T ny(n)) does not always converge.