2025/08/08 by Faruk Alpay, Alpay, Faruk, Taylan Alpay +3 · 1 citation
Computer Science · Mathematics · #47A10 #47A15 #47A35 #47A60 #47B15 #47D06 #FOS: Mathematics #Functional Analysis (math.FA) #G.1.0 #G.1.10 #G.1.3 #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2508.06025
openalex publication_date 2025/08/08 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28
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.