About the sharpness of the stability estimates in the Kreiss matrix theorem

Abstract

Abstract. One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices A under consideration. This so-called resolvent condition is known to imply, for all n ≥ 1, the upper bounds A n ≤ eK(N + 1) and A n ≤ eK(n + 1). Here · is the spectral norm, K is the constant occurring in the resolvent condition, and the order of A is equal to N + 1 ≥ 1.It is a long-standing problem whether these upper bounds can be sharpened, for all fixed K > 1, to bounds in which the right-hand members grow much slower than linearly with N + 1 and with n + 1, respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each > 0, there are fixed values C > 0, K > 1 and a sequence of (N + 1) × (N + 1) matrices A N , satisfying the resolvent condition, such thatThe result proved in this paper is also relevant to matrices A whose -pseudospectra lie at a distance not exceeding K from the unit disk for all > 0.

Functional calculus under the Tadmor–Ritt condition, and free interpolation by polynomials of a given degree

“…(For example, on X ¼ L 1 ðmÞ; L N ðmÞ for every measure m not reduced to a finite sum of d-measures, or on CðKÞ; where K is an infinite compact.) On a Hilbert space, using results from [STW03], we can prove the following.…”

“…On the other hand, UBðE N ÞpNBðEÞ þ 1pNð1 þ bÞ þ 1: & For Hilbert spaces, we have only the following lemma which is mostly known; property (1) is proved in McCarthy and Schwartz [MS65] and property (2) in Spijker et al [STW03].…”

“…Upper estimates for the polynomial functional calculus of Kreiss operators on specific Banach spaces, which, in principle, could be better than the general one, are not known (for instance, for Hilbert spaces). As to the lower ones, Spijker et al [STW03] constructed operators T e ; e40; on a Hilbert space, satisfying ðKÞ and such that C T ðnÞXjjT n jjXa e ðn þ 1Þ 1Àe (where a e -0 as e-0). See also Remark 2.13.…”

Functional calculus under Kreiss type conditions

“…Here R λ (T ) = (λI − T ) −1 is the resolvent of T . The bound is known to be sharp, see [9] (and [20] for the Hilbert space case). For the case dim X = ∞, the same resolvent condition only implies that T n ≤ Cen for n ∈ N. Many other resolvent conditions were considered in order to get estimates for T n , n ≥ 0.…”

“…The latter are of interest for numerical analysis since they appear when applying iteration methods for solving partial differential equations. For the corresponding results, references and the history of the problem we refer to [8], [12], [16], [20] and [21].…”