Transfinite Iteration of Operator Transforms and Spectral Projections in Hilbert and Banach Spaces

TL;DR

This paper investigates the convergence properties of transfinite iterations of operator transforms in Hilbert and Banach spaces, establishing conditions under which these iterations approach spectral and ergodic projections.

Contribution

It introduces a framework for transfinite iteration of operator transforms, proving convergence to spectral projections under specific functional calculus conditions.

Findings

Convergence of iterates to spectral projections in Hilbert spaces.

Identification of spectra via essential range at finite stages.

Explicit construction of Schur filters and counterexamples.

Abstract

We study ordinal-indexed, multi-layer iterations of bounded operator transforms and prove convergence to spectral/ergodic projections under functional-calculus hypotheses. For normal operators on Hilbert space and polynomial or holomorphic layers that are contractive on the spectrum and fix the peripheral spectrum only at fixed points, the iterates converge in the strong operator topology by a countable stage to the spectral projection onto the joint peripheral fixed set. We describe spectral mapping at finite stages and identify the spectrum of the limit via the essential range. In reflexive Banach spaces, for Ritt or sectorial operators with a bounded H-infinity functional calculus, the composite layer is power-bounded and its mean-ergodic projection yields an idempotent commuting with the original operator; under a peripheral-separation condition the powers converge strongly to this projection. We provide explicit two-layer Schur filters, a concise Schur/Nevanlinna-Pick lemma, a Fejer-type monotonicity bound implying stabilization by the first countable limit (omega), examples that attain exactly the omega stage, and counterexamples outside the hypotheses.