Iterates of Ritt operators close to the identity

Abstract

We study bounded operators whose prescribed powers remain uniformly close to the identity, and resolvent conditions sampled along sequences of powers. The displacement bound ${\sup }_{n\ge 1}\Vert I-T^n\Vert \le q<2$ forces a finite peripheral spectrum consisting of odd-order roots of unity and makes an explicitly determined odd power of $T$ a Ritt operator. The sharp unconditional threshold for $T$ itself is $\sqrt{3}$. If ${\Phi }_{\ast }(q)$ denotes the optimal universal Ritt resolvent bound below this threshold, then ${\Phi }_{\ast }(q)=2/(\sqrt{3}-q)+O(1)$ as $q\uparrow \sqrt{3}$. At every higher peripheral threshold the optimal finite-peripheral resolvent bound has reciprocal order of growth. We also obtain constructive bounds for general weighted Wiener symbols. For each $1<q<2$, an angular escape invariant characterises exactly the sampling sequences for which a displacement bound by $q$, together with the peripheral spectral condition, forces the Ritt property. We compute this invariant for asymptotically geometric sequences. At the endpoint $q=1$, suitable phase conditions imply the full displacement bound without assuming power boundedness; normal contractions admit an exact scalar criterion. Finally, we establish quantitative Hilbert-space and $L^p$ resolvent estimates, a characterisation of operators with odd Ritt powers on uniformly convex spaces, and strict norm gaps for operator-valued disc-algebra functions, with explicit matrix-polynomial bounds from Fejér--Riesz factorisation.

45 pages