A weighted composition semigroup related to three open problems

2021/11/09 by Juan Manzur, Manzur, Juan, S. Waleed Noor +3 · 2 citations
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical functions and polynomials #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2111.05398

openalex publication_date 2021/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The semi-group of weighted composition operators (W_n)_{n≥ 1} where W_nf(z)=(1+z+…+z^n-1)f(zⁿ) on the classical Hardy-Hilbert space H² of the open unit disk is related to the Riemann Hypothesis (RH) (see \citeWaleed). The semigroup (W_n)_{n≥ 1} is also closely related to the Invariant Subspace Problem (ISP) and the Periodic Dilation Completeness Problem (PDCP). We obtain results on cyclic vectors, spectra, invariant and reducing subspaces. In particular, we show that several basic questions related to the semigroup (W_n)_{n≥ 1} are equivalent to the RH and provide generalizations of the Báez-Duarte criterion for the RH (see \citeBaez-Duarte).

Cited by

• Recent perspectives on the Invariant Subspace Problem
• On the dynamics of a semigroup and its relation with the Riemann Hypothesis

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