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Quantifying (non-)weak compactness of operators on $AL$- and $C(K)$-spaces

Antonio Acuaviva, Amir Bahman Nasseri

TL;DR

The paper develops a unified quantitative framework for non-weakly compact operators on AL- and C(K)-spaces by establishing best weakly compact approximants, relating the weak essential norm to localized obstructions on finite-measure subsets, and proving a general factorization through $oldsymbol{ m ell}_1$ (and dual through $c_0$ or $oldsymbol{ m eta b N}$) that yields a unique algebra norm on the weak Calkin algebra. The method hinges on decomposing operators into integral and singular parts, leveraging Weis's representations, lattice-isomorphism transport, and rich families to extend finite-measure results to strictly localizable spaces. Consequently, the work provides dual results for $C(K)$-spaces (via extremally disconnected $L$) and resolves questions about equivalences among weak-noncompactness measures (weak essential norm, residuum, and De Blasi measure) in key cases, including $T: L_ olinebreak_ olinebreakinfty[0,1] o L_ olinebreak_ olinebreakinfty[0,1]$. It also clarifies when a quantitative factorization is possible and when it fails, clarifying structural properties of AL- and C(K)-spaces and their weak Calkin algebras with significant implications for operator ideals and norm topology.

Abstract

We study the representation of non-weakly compact operators between $AL$-spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between $C(L)$-spaces where $L$ is extremally disconnected. We also characterize the weak essential norm for operators between $AL$-spaces in terms of factorizations of the identity on $\ell_1$. As a consequence, we deduce that the weak Calkin algebra $\mathscr{B}(E)/\mathscr{W}(E)$ admits a unique algebra norm for every $AL$-space $E$. By duality, similar results are obtained for $C(K)$-spaces. In particular, we prove that for operators $T: L_{\infty}[0,1] \to L_{\infty}[0,1]$ the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.

Quantifying (non-)weak compactness of operators on $AL$- and $C(K)$-spaces

TL;DR

The paper develops a unified quantitative framework for non-weakly compact operators on AL- and C(K)-spaces by establishing best weakly compact approximants, relating the weak essential norm to localized obstructions on finite-measure subsets, and proving a general factorization through (and dual through or ) that yields a unique algebra norm on the weak Calkin algebra. The method hinges on decomposing operators into integral and singular parts, leveraging Weis's representations, lattice-isomorphism transport, and rich families to extend finite-measure results to strictly localizable spaces. Consequently, the work provides dual results for -spaces (via extremally disconnected ) and resolves questions about equivalences among weak-noncompactness measures (weak essential norm, residuum, and De Blasi measure) in key cases, including . It also clarifies when a quantitative factorization is possible and when it fails, clarifying structural properties of AL- and C(K)-spaces and their weak Calkin algebras with significant implications for operator ideals and norm topology.

Abstract

We study the representation of non-weakly compact operators between -spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between -spaces where is extremally disconnected. We also characterize the weak essential norm for operators between -spaces in terms of factorizations of the identity on . As a consequence, we deduce that the weak Calkin algebra admits a unique algebra norm for every -space . By duality, similar results are obtained for -spaces. In particular, we prove that for operators the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.

Paper Structure

This paper contains 15 sections, 49 theorems, 215 equations.

Table of Contents

  1. Introduction and Main Results
  2. Organization and notation
  3. Preliminaries
  4. Aspects about Riesz spaces.
  5. Measure theoretic aspects.
  6. Integral operators and singular operators on $\sigma$-finite $L_1$-spaces.
  7. Weak compactness in $L_1$-spaces.
  8. The proof of Theorem \ref{['th: MainTheorem']}
  9. Standard measure spaces.
  10. Measure spaces of finite measure with countably generated $\sigma$-algebra.
  11. Measure spaces of finite measure.
  12. Strictly localizable measure spaces.
  13. The Weak Calkin Algebra in $AL$-spaces
  14. Quantitative weak noncompactness of operators on $AL$-spaces.
  15. The Weak Calkin Algebra in $C(K)$-spaces

Key Result

Theorem 1.1

Let $(X, \Sigma, \mu)$ and $(Y, \Gamma, \nu)$ be strictly localizable measure spaces, and $T: L_1(X, \Sigma, \mu) \to L_1(Y, \Gamma, \nu)$ be an operator. Then there exists a weakly compact operator $W: L_1(X, \Sigma, \mu) \to L_1(Y, \Gamma, \nu)$ such that where $\alpha_\Theta (T) = \limsup_{\nu(B) \to 0, B \subseteq \Theta} \lVert\chi_{Y \backslash \Theta} T + \chi_B T\rVert$.

Theorems & Definitions (112)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.11
  • ...and 102 more