Problems / analysis

analysis / Operator theory

Separation Between the Ordinary and Strong Kreiss Constants

Question 6.1 of Chalmoukis, Tsikalas and Yakubovich asks how far the Power boundedness constant $P(T)$ of a matrix can exceed its ordinary Kreiss constant $K(T)$. Answered more strongly: for every $K > 1$ there are matrices whose Cayley transforms satisfy $K(C_h(A_{n,h})) \le K$ while the strong Kreiss constant satisfies $K_s(C_h(A_{n,h})) \ge \tfrac{1}{2}Cn^{\alpha_K}$ with $\alpha_K = (K-1)/(C+K-1)$. Since $P(T)\geq K_s(T)$, this solves the question. Moreover, since the Kreiss matrix theorem gives $K_s(T) \le P(T) \le edK(T)$ in dimension $d$, the exponent $\alpha < 1$ is optimal up to an arbitrarily small power loss.