Searcharxiv⌕ Search

arXiv subjects

S. Waleed Noor

Publications and source records attributed to S. Waleed Noor.

12 recordsLinked to original sources
arXiv2024

Beurling and Model subspaces invariant under a universal operator

In this article, we characterize the Beurling and Model subspaces of the Hardy-Hilbert space $H^2(\mathbb{D})$ invariant under the composition operator $C_{ϕ_a}f=f\circϕ_a$, where $ϕ_a(z) = az + 1 - a$ for $a \in (0,1)$ is an affine self-map of the open unit disk $\mathbb{D}$. These operators have universal translates (in the sense of Rota) and have attracted attention recently due to their connection with the Invariant Subspace Problem (ISP) and the classical Cesàro operator.

math.FA↗
arXiv2024

Zero-free half-planes of the ζ-function via spaces of analytic functions

In this article, we introduce a general approach for deriving zero-free half-planes for the Riemann zeta function $ζ$ by identifying topological vector spaces of analytic functions with specific properties. This approach is applied to weighted $\ell^2$ spaces and classical Hardy spaces $ H^p $ ($ 0<p\leq2 $). As a consequence precise conditions are obtained for the existence of zero-free half planes for the $ζ$-function.

math.NT↗
arXiv2023

Minimal invariant subspaces for an affine composition operator

The composition operator $C_{ϕ_a}f=f\circϕ_a$ on the Hardy-Hilbert space $H^2(\mathbb{D})$ with affine symbol $ϕ_a(z)=az+1-a$ and $0<a<1$ has the property that the Invariant Subspace Problem for complex separable Hilbert spaces holds if and only if every minimal invariant subspace for $C_{ϕ_a}$ is one-dimensional. These minimal invariant subspaces are always singly-generated $ K_f := \overline{\mathrm{span} \{f, C_{ϕ_a}f, C^2_{ϕ_a}f, \ldots \}}$ for some $f\in H^2(\mathbb{D})$. In this article we characterize the minimal $K_f$ when $f$ has a nonzero limit at the point $1$ or if its derivative $f'$ is bounded near $1$. We also consider the role of the zero set of $f$ in determining $K_f$. Finally we prove a result linking universality in the sense of Rota with cyclicity.

math.FA↗
arXiv2021

Universal composition operators

A Hilbert space operator $U$ is called universal (in the sense of Rota) if every Hilbert space operator is similar to a multiple of $U$ restricted to one of its invariant subspaces. It follows that the Invariant Subspace Problem for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for $U$ are one dimensional. In this article we characterize all linear fractional composition operators $C_ϕ f=f\circϕ$ that have universal translates on both the classical Hardy spaces $H^2(\mathbb{C}_+)$ and $H^2(\mathbb{D})$ of the half-plane and the unit disk respectively. The surprising new example is the composition operator on $H^2(\mathbb{D})$ with affine symbol $ϕ_a(z)=az+(1-a)$ for $0<a<1$. This leads to strong characterizations of minimal invariant subspaces and eigenvectors of $C_{ϕ_a}$ and offers an alternative approach to the ISP.

math.FA↗
arXiv2020

Interplay between complex symmetry and Koenigs eigenfunctions

We investigate the relationship between the complex symmetry of composition operators $C_ϕf=f\circ ϕ$ induced on the classical Hardy space $H^2(\mathbb{D})$ by an analytic self-map $ϕ$ of the open unit disk $\mathbb{D}$ and its Koenigs eigenfunction. A generalization of orthogonality known as conjugate-orthogonality will play a key role in this work. We show that if $ϕ$ is a Schröder map (fixes a point $a\in \mathbb{D}$ with $0<|ϕ'(a)|<1$) and $σ$ is its Koenigs eigenfunction, then $C_ϕ$ is complex symmetric if and only if $(σ^n)_{n\in \mathbb{N}}$ is complete and conjugate-orthogonal in $H^2(\mathbb{D})$. We study the conjugate-orthogonality of Koenigs sequences with some concrete examples. We use these results to show that commutants of complex symmetric composition operators with Schröder symbols consist entirely of complex symmetric operators.

math.FA↗
arXiv2020

Universal Toeplitz operators on the Hardy space over the polydisk

The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota), the ISP may be solved by describing the invariant subspaces of these operators alone. We characterize all anaytic Toeplitz operators $T_ϕ$ on the Hardy space $H^2(\mathbb{D}^n)$ over the polydisk $\mathbb{D}^n$ for $n>1$ whose adjoints satisfy the Caradus criterion for universality, that is, when $T_ϕ^*$ is surjective and has infinite dimensional kernel. In particular if $ϕ$ in a non-constant inner function on $\mathbb{D}^n$, or a polynomial in the ring $\mathbb{C}[z_1,\ldots,z_n]$ that has zeros in $\mathbb{D}^n$ but is zero-free on $\mathbb{T}^n$, then $T_ϕ^*$ is universal for $H^2(\mathbb{D}^n)$. The analogs of these results for $n=1$ are not true.

math.FA↗
arXiv2019

Complex symmetry and cyclicity of composition operators on $H^2(\mathbb{C}_+)$

In this article, we completely characterize the complex symmetry, cyclicity and hypercyclicity of composition operators $C_ϕf=f\circϕ$ induced by affine self-maps $ϕ$ of the right half-plane $\mathbb{C}_+$ on the Hardy-Hilbert space $H^2(\mathbb{C}_+)$. We also provide new proofs for the normal, self-adjoint and unitary cases and for an adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodrígues.

math.FA↗
arXiv2018

Complex symmetric composition operators

Let $φ$ be a linear fractional self-map of the open unit disk $\mathbb{D}$ and $H^2$ the Hardy space of analytic functions on $\mathbb{D}$. The goal of this article is to characterize the linear fractional composition operators $C_φf=f\circφ$ on $H^2$ that are complex symmetric.

math.FA↗
arXiv2012

Complex symmetry of Composition operators induced by involutive Ball automorphisms

Suppose $\mathcal{H}$ is a weighted Hardy space of analytic functions on the unit ball $\mathbb{B}_n\subset\mathbb{C}^n$ such that the composition operator $C_ψ$ defined by $C_ψf=f\circψ$ is bounded on $\mathcal{H}$ whenever $ψ$ is a linear fractional self-map of $\mathbb{B}_n$. If $φ$ is an involutive Moebius automorphism of $\mathbb{B}_n$, we find a conjugation operator $\mathcal{J}$ on $\mathcal{H}$ such that $C_φ=\mathcal{J} C^*_φ\mathcal{J}$. The case $n=1$ answers a question of Garcia and Hammond.

math.FA↗
arXiv2012

Embeddings of Müntz Spaces: Composition Operators

Given a strictly increasing sequence $Λ=(λ_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{λ_n}<\infty$, the Müntz spaces $M_Λ^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{λ_n}$. We discuss how properties of the embedding $M_Λ^2\subset L^2(μ)$, where $μ$ is a finite positive Borel measure on the interval $[0,1]$, have immediate consequences for composition operators on $M^2_Λ$. We give criteria for composition operators to be bounded, compact, or to belong to the Schatten--von Neumann ideals.

math.FA↗
arXiv2011

Embeddings of Müntz spaces: the Hilbertian case

Given a strictly increasing sequence $Λ=(λ_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{λ_n}<\infty$, the Müntz spaces $M_Λ^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{λ_n}$. We discuss properties of the embedding $M_Λ^p\subset L^p(μ)$, where $μ$ is a finite positive Borel measure on the interval $[0,1]$. Most of the results are obtained for the Hilbertian case $p=2$, in which we give conditions for the embedding to be bounded, compact, or to belong to the Schatten--von Neumann ideals.

math.FA↗