Sublinear dimension growth in the Kreiss Matrix Theorem

Abstract

A possible sublinear dimension growth in the Kreiss Matrix Theorem, bounding the stability constant in terms of the Kreiss resolvent characteristic, is discussed. Such a growth is proved for matrices having unimodular spectrum and acting on a uniformly convex Banach space. The principal ingredients to results obtained come from geometric properties of eigenvectors, where the approaches by C. A. McCarthy-J. Schwartz (1965) and V. I. Gurarii-N. I. Gurarii (1971) are used and compared. The sharpness issue is verified via finite Muckenhoupt bases (by using mostly the approach by M. Spijker, S. Tracogna, and B. Welfert (2003)).

On measuring unboundedness of the H∞-calculus for generators of analytic semigroups

“…The approach moreover relies on estimates of derivatives of the (analytic) semigroup. This results in a similar estimate as in (2.11), but with the following constants C1 = 30 π M 2 , C2 = 16 π M 3 , C3 = 30 π M 3 . Thus, by our results, the M -dependence gets improved from M 3 to M (1 + log M ).…”

“…For general Banach spaces this does not hold. However, there exists a version of McCarthy-Schwartz's result for uniformly convex spaces by Gurarii and Gurarii [15], see also [30,Thm. 3.6.1 and Cor.…”

“…Therefore, for fixed M (A, φ), the behavior in ε → 0 + could be better than | log ε|. For a similar effect we refer to the question of sharpness of Spijker's result on the Kreiss-Matrix-Theorem, see [37,38] and the recent contribution by Nikolski [30]…”

Sharp growth rates for semigroups using resolvent bounds

“…Proposition 3.14 is sharp in the case where g(t) = Kt for some K ≥ 0 and all t > 0 (see [26,31,44]). For further discussion on this topic we refer the reader to [35], where in particular improvements on the bounds have been obtained under additional geometric assumptions on the norm of X.…”

Functional calculus estimates for Tadmor–Ritt operators

“…For the detailed history of the result we refer to the monograph [47] and the recent work [37]. By [29], estimate (1.5) is sharp in the sense that there exists a sequence of matrices T N ∈ KR(C N×N ) such that lim N→∞ Pb(T N ) C Kreiss (T N )N = e.…”