[Paper Review] Inequalities for Functions of Selfadjoint Operators on Hilbert Spaces

Sever S Dragomir|arXiv (Cornell University)|Mar 8, 2012

Mathematical Inequalities and Applications27 references19 citations

TL;DR

This paper establishes sharp error bounds for Taylor-type approximations of functions of selfadjoint operators on Hilbert spaces, leveraging spectral theory and integral representations. It derives explicit inequalities for the remainder term in Taylor expansions using absolute continuity and integrability conditions on higher-order derivatives, with key results expressed via Lebesgue norms and spectral measures.

ABSTRACT

The main aim of this book is to present recent results concerning inequalities for continuous functions of selfadjoint operators on complex Hilbert spaces. It is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas.

Motivation & Objective

• To develop precise error estimates for Taylor-type expansions of functions applied to selfadjoint operators on Hilbert spaces.
• To extend classical Taylor remainder bounds to the operator setting using spectral decomposition and integral representations.
• To quantify the remainder term in terms of the total variation and Lp norms of higher-order derivatives.
• To provide computable bounds under absolute continuity and integrability assumptions on the n-th derivative.
• To unify and generalize existing inequalities in operator theory, particularly for convex, log-convex, and Lipschitzian functions.

Proposed method

• Utilizes the spectral decomposition of selfadjoint operators to represent f(A) via a Stieltjes integral with respect to the spectral family {Eλ}.
• Derives an integral representation of the Taylor remainder Tn(f, m, M; x, y) involving a kernel Wn(m, M, f; λ) defined on [m, M].
• Applies Hölder's inequality to bound |Wn(m, M, f; λ)| in terms of Lp norms of f(n+1) over subintervals [m, λ] and [λ, M].
• Establishes bounds for |Tn(f, m, M; x, y)| by integrating |Wn| against the spectral measure ⟨Eλx, y⟩, leading to operator norm estimates.
• Considers multiple cases based on the integrability class of f(n+1), including L∞, Lq, and L1, yielding 12 distinct bound variants.
• Applies the Total Variation Schwarz inequality to refine the final error bound in terms of ∥x∥∥y∥ and the L1 norm of |Wn|.

Experimental results

Research questions

• RQ1 What are the optimal bounds for the remainder term in a Taylor expansion of a function applied to a selfadjoint operator?
• RQ2 How can the remainder be expressed and bounded using the spectral measure and higher-order derivatives?
• RQ3 What are the error bounds when f(n) is absolutely continuous and f(n+1) belongs to Lp spaces?
• RQ4 How do the bounds depend on the smoothness and variation properties of f(n+1)?
• RQ5 Can the remainder be uniformly bounded in terms of operator norms and spectral integral norms?

Key findings

• The remainder Tn(f, m, M; x, y) admits the integral representation Tn = 1/((M−m)n!) ∫_m^M Wn(λ) ⟨Eλx, y⟩ dλ, where Wn is a kernel function involving f(n+1).
• For f(n+1) ∈ L∞[m,M], the remainder satisfies |Tn| ≤ [(M−m)n+1 / 4(n+1)!] Ln ∥x∥∥y∥, where Ln is the L∞-seminorm of f(n+1).
• When f(n+1) ∈ L∞[m,λ] and [λ,M], the kernel |Wn(λ)| is bounded by a sum of four terms involving (λ−m), (M−λ), and norms of f(n+1).
• The bound |Tn(f, m, M; x, y)| ≤ 1/((M−m)n!) ∫_m^M |Wn(λ)| ⟨Eλx, x⟩^{1/2} ⟨Eλy, y⟩^{1/2} dλ holds for all x, y ∈ H.
• A sharp bound is obtained as |Tn| ≤ 1/((M−m)n!) ∥x∥∥y∥ ∫_m^M |Wn(λ)| dλ, with |Wn(λ)| ≤ ∑_{i=1}^4 B_n^{(i)}(λ) under various Lp assumptions.
• The result generalizes classical Taylor error bounds to the non-commutative operator setting using spectral theory and integral inequalities.