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

For a bounded linear operator T:H→H with dense range on the Hilbert space H,x0set membership, variantH and short parallelx0short parallel>ε>0, the backward minimal point y(n) is the unique vector of smallest norm in the set {y:short parallelTny−x0short parallelless-than-or-equals, slantε}. We investigate the limit of the sequence (Tny(n)) for operators T on finite-dimensional vector spaces. This vector—lim Tny(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 image and self-adjoint operators on image with orthogonal eigenvectors. We also show that the sequence (Tny(n)) does not always converge.