On GMRES-equivalent bounded operators

Given a bounded linear operator A in a Hilbert space H and a nonzero vector r is an element of H, we construct a unitary operator U and (under some co nditions) bounded self-adjoint operators P and T (nonnegative definite and indefinite, respectively) such that all the residual Krylov subspaces of (A , r), (U, r), (P, r), and (T, r) of the same dimension for the equation Ax = r are equal. When possible (for example, for U and P, provided 0 is outsi de the field of values of A), we estimate a gap in the spectrum of U and th e condition numbers of P and T. Some attainability results are also establi shed. It is shown that some analogous matrix assertions are valid, which can be o btained by means of degenerating the operator case. Numerical examples are presented for the finite-dimensional case.