The Faber-Manteuffel Theorem for Linear Operators

A short recurrence for orthogonalizing Krylov subspace bases for a matrix A exists if and only if the adjoint of A is a low degree polynomial in A (i.e. A is normal of low degree). In the area of iterative methods, this result is known as the Faber-Manteuffel Theorem (V. Faber and T. Manteuffel, SIAM J. Numer. Anal., 21 (1984), pp. 352–362). Motivated by the description in (J. Liesen and Z. Strakoš, On optimal short recurrences for generating orthogonal Krylov subspace bases, SIAM Rev., to appear), we here formulate this theorem in terms of linear operators on finite dimensional Hilbert spaces, and give two new proofs of the necessity part. We have chosen the linear operator rather than the matrix formulation because we found that a matrix-free proof is less technical. Of course, the linear operator result contains the Faber-Manteuffel Theorem for matrices.