Constants related to operators of class C_ρ

Abstract

Let T be a Hilbert-space operator of class

$$\left\| {S^{ - 1} TS} \right\| \leqslant 1\,\,\,\,and\,\,\left\| {S^{ - 1} } \right\| \cdot \left\| S \right\| \leqslant \max (1,\rho )$$

in the sense of Sz.-Nagy and Foiaş. Then there exists an invertible operator S such that ∥S⁻¹TS∥≤1 and ∥S⁻¹∥·∥S∥≤max(1,ρ). The following estimate is valid for the bilateral limit:

$$\mathop {lim}\limits_{n \to \infty } \{ \left\| {T^n h} \right\|^2 + \left\| {T*^n h} \right\|^2 \} \leqslant max(2,\rho )\left\| h \right\|^2 .$$

Here the constants max(1,ρ) and max(2,ρ) are the best possible in their respective cases.