the preconditioned matrix [C.sup.-1] A is

diagonalizable by a similarity transformation formulation, see e.g.

In the Hermitian

diagonalizable (i.e., physical) cases, these EP values are all necessarily complex so that whenever the parameter remains real, the distances between the separate real eigenvalues behave as if controlled by a mutual "repulsion" (A deeper analysis may be found in the Kato's book [33]).

For simplicity of notation, we suppose that A has n-different eigenvalues [[gamma].sub.1], [[gamma].sub.2], ..., [[gamma].sub.n] (in this case, A is

diagonalizable).

Dupain, "Estimates for the Bergman and Szego projections for pseudoconvex domains of finite type with locally

diagonalizable Levi form," Publicacions Matematiques, vol.

In fact, such operators are

diagonalizable. In [8], the numerical ranges of such operators on Hardy space of D are studied.

Among his topics are representations of solutions to operator equations, bounds for condition numbers of

diagonalizable matrices, functions of a compact operator in a Hilbert space, regular functions of a bounded non-self-adjoint operator, and commutators and perturbations of operator functions.

Since A is an Anosov matrix, A is

diagonalizable. So, there exists P [member of] GL (2, R) such that [mathematical expression not reproducible], where [[lambda].sub.1] and [[lambda].sub.2] are the eigenvalues of A.

These expressions show precisely how the residual norms depend on the eigenvalues, the pairwise differences of eigenvalues and on the eigenvectors (or principal vectors if A is not

diagonalizable).

We just observe that, apart from the above calculation, studying the sample properties of the truncated spectrum is made hard by the fact that eigenvalues and eigenvectors of a

diagonalizable matrix are intimately related from their very definition; thus such study would require a careful consideration of the distribution of the sample diagonalizing matrix [mathematical expression not reproducible].

When the matrix T is

diagonalizable and (I - T) is nonsingular, vector sequences {[x.sub.n]} satisfy the conditions of Theorem 1 [17].