[Paper Review] Bennett-Carl inequalities for symmetric Banach sequence spaces and unitary ideals

Andreas Defant, Carsten Michels|ArXiv.org|Apr 28, 1999

Advanced Banach Space Theory25 references3 citations

TL;DR

This paper establishes abstract Bennett-Carl inequalities for operators between symmetric Banach sequence spaces and unitary ideals by introducing a novel interpolation theorem that bridges (r,2)-summing and (s,2)-mixing norms. The key contribution is a unified framework that recovers classical Bennett-Carl inequalities for Minkowski spaces ℓᵤ and Schatten classes Sᵤ, while extending them to broader classes of symmetric sequence spaces and unitary ideals via interpolation techniques.

ABSTRACT

We prove an abstract interpolation theorem which interpolates the (r,2)-summing and (s,2)-mixing norm of a fixed operator in the image and the range space. Combined with interpolation formulas for spaces of operators we obtain as an application the original Bennett-Carl inequalities for identities acting between Minkowski spaces l_u as well as their analogues for Schatten classes S_u. Furthermore, our techniques motivate a study of Bennett-Carl inequalities within a more general setting of symmetric Banach sequence spaces and unitary ideals.

Motivation & Objective

• To develop a general interpolation framework for operator norms in symmetric Banach sequence spaces.
• To unify and generalize the classical Bennett-Carl inequalities for ℓᵤ spaces and Schatten classes Sᵤ.
• To extend the theory of (r,2)-summing and (s,2)-mixing norms to the broader context of symmetric Banach sequence spaces and unitary ideals.
• To establish a theoretical foundation for interpolation theorems applicable to operator ideals beyond classical Lorentz and Schatten classes.
• To provide a systematic approach to deriving Bennett-Carl type inequalities in abstract symmetric sequence space settings.

Proposed method

• An abstract interpolation theorem is formulated to interpolate between (r,2)-summing and (s,2)-mixing norms of a fixed operator.
• The method relies on interpolation formulas for spaces of operators, particularly in the context of symmetric Banach sequence spaces.
• The framework is applied to derive the original Bennett-Carl inequalities for identity operators on Minkowski sequence spaces ℓᵤ.
• The same techniques are extended to unitary ideals, particularly Schatten classes Sᵤ, yielding analogues of Bennett-Carl inequalities.
• The approach uses duality and interpolation theory to generalize results beyond classical Lorentz and Schatten spaces.
• Theoretical tools from functional analysis, including operator ideals and symmetric norms, are systematically employed.

Experimental results

Research questions

• RQ1 How can the classical Bennett-Carl inequalities for ℓᵤ and Sᵤ be derived from a unified interpolation principle?
• RQ2 What is the role of symmetric Banach sequence spaces in generalizing Bennett-Carl inequalities beyond classical Lorentz and Schatten classes?
• RQ3 Can interpolation theorems for operator norms be extended to unitary ideals using abstract symmetric sequence space structures?
• RQ4 What conditions ensure that (r,2)-summing and (s,2)-mixing norms interpolate consistently across symmetric sequence spaces?
• RQ5 How do the new inequalities relate to known interpolation formulas in the theory of operator ideals?

Key findings

• The paper establishes a general interpolation theorem that bridges (r,2)-summing and (s,2)-mixing norms for operators between symmetric Banach sequence spaces.
• The method successfully recovers the original Bennett-Carl inequalities for identity operators on Minkowski spaces ℓᵤ.
• Analogous Bennett-Carl inequalities are derived for Schatten classes Sᵤ, extending known results to unitary ideals.
• The framework generalizes to arbitrary symmetric Banach sequence spaces, providing a broader setting for such inequalities.
• The results are achieved through systematic use of interpolation theory and duality in operator ideals.
• The study reveals that symmetric structure in sequence spaces and unitary ideals enables a unified treatment of Bennett-Carl type inequalities.