A weighted composition semigroup related to three open problems

TL;DR

This paper studies a specific weighted composition semigroup on the Hardy space, revealing connections to the Riemann Hypothesis, invariant subspace problem, and periodic dilation completeness problem, with new results on spectra and invariant subspaces.

Contribution

It establishes new links between the semigroup's properties and major open problems, generalizes criteria related to the Riemann Hypothesis, and analyzes invariant structures.

Findings

Several questions about the semigroup are equivalent to the Riemann Hypothesis.

Provides new criteria generalizing Baez-Duarte's criterion for RH.

Results on spectra, cyclic vectors, and invariant subspaces of the semigroup.

Abstract

The semi-group of weighted composition operators $(W_n)_{n\geq 1}$ where \[ W_nf(z)=(1+z+\ldots+z^{n-1})f(z^n) \] on the classical Hardy-Hilbert space $H^2$ of the open unit disk is related to the Riemann Hypothesis (RH) (see \cite{Waleed}). The semigroup $(W_n)_{n\geq 1}$ 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 $(W_n)_{n\geq 1}$ are equivalent to the RH and provide generalizations of the B\'aez-Duarte criterion for the RH (see \cite{Baez-Duarte}).