[Paper Review] Norm of the discrete Cesàro operator minus identity

Gord Sinnamon|arXiv (Cornell University)|May 29, 2021

Holomorphic and Operator Theory3 references26 citations

TL;DR

The paper exactly determines the operator norm of C−I on ℓ^p for all p∈(1,∞): it equals 1/(p−1) for 1<p≤2 and m_p^{−1/p} for p>2 (with p=∞ giving 2); it also aligns with the continuous Hardy case and introduces m_p via a minimization of a related function.

ABSTRACT

The norm of $C-I$ on $\ell^p$, where $C$ is the Cesàro operator, is shown to be $1/(p-1)$ when $1

Motivation & Objective

• Motivate and formalize Bennett’s 1996 question on the norm of C−I on ℓ^p and Jameson’s conjecture.
• Extend known results from p=4/3 and p=2 to all p>1, providing exact norms for all p.
• Relate discrete ℓ^p results to the continuous Hardy operator P and its transpose, showing common norms across discrete and continuous settings.
• Develop a unified framework using transpose operators, Hölder inequality, and duality to establish sharp bounds and exact values.

Proposed method

• Define the transpose Cesàro operator C^T and use the duality ∥C−I∥_ℓ^p = ∥C^T−I∥_ℓ^{p′} to study the range p>2.
• Prove a key inequality via Lemma 1 that yields an upper bound ∥C^T−I∥_ℓ^p ≤ p−1 for p in a dense set 𝔼 and extend by continuity.
• Apply Hölder’s inequality and summation tricks to derive a finite bound ∑(y_n−x_n)^p ≤ (p−1)^p ∑x_n^p (and its continuous analogue).
• Introduce the function f_p(t)=p t^{p−1}+(1−t)^p−t^p and its minimum m_p on [0,1/2], showing m_p is attained at a unique t_p in (0,1/2) for p>2.
• Employ the Riesz–Thorin interpolation between p-values in 𝔼 to extend bounds to all p>2, and use a constructive extremal sequence to show sharpness, yielding ∥C−I∥_ℓ^p = m_p^{−1/p} for p>2.
• Demonstrate the discrete results parallel the continuous case with P^T and P, and derive corresponding L^p norms for P−I via analogous arguments.

Experimental results

Research questions

• RQ1 What is the exact norm of C−I on ℓ^p for 1<p<∞?
• RQ2 Does the known p=4/3 case (and related conjectures) extend to all p in (1,∞) and how?
• RQ3 How do discrete ℓ^p results compare to the continuous Hardy operator case P−I, and can parallels be made precise?
• RQ4 What role does the auxiliary function m_p play in determining the sharp norm for p>2?

Key findings

• For 1<p≤2, ∥C−I∥_ℓ^p = 1/(p−1) (equivalently ∥C^T−I∥_ℓ^p = p−1).
• For p>2, ∥C−I∥_ℓ^p = m_p^{−1/p}, where m_p is the minimum of f_p(t)=p t^{p−1}+(1−t)^p−t^p on [0,1/2].
• In particular, p=∞ yields ∥C−I∥_ℓ^∞ = 2.
• The results extend Jameson’s bound for p=4 and recover his exact value for p=4/3 and p=3 through specific computations of m_p and t_p.
• The discrete results mirror the continuous case: ∥P−I∥_{L^p} equals the same expressions, and corollaries provide parallel statements for P^T and the positive cone cases.
• The proofs combine Hölder’s inequality, a key inequality (Lemma 1), Riesz–Thorin interpolation, and explicit extremal constructions to show sharpness.