Left-hand Quotients Lipschitz Ideals by Operator Ideals

Document Type : Original Article

Authors

Amar Belacel ¹

Amar Bougoutaia ¹

Antonio Jim\'{e}nez-Vargas ²

¹ Laboratory of Pure and Applied Mathematics (LPAM)‎, ‎University of Laghouat‎, ‎Laghouat‎, ‎Algeria

² Universidad de Almería Carretera Sacramento s/n 04120 La Cañada de San Urbano, Almería, Spain

10.22067/kjm.2025.551946.3698

Abstract

 Expanding on the concept of left-hand quotients of operator ideals, we analyze the known notion of Lipschitz left-hand quotient $A^{-1} \circ I_{\mathrm{Lip}}$, where $A$ denotes an operator ideal and $I_{\mathrm{Lip}}$ represents a Lipschitz ideal. We establish that these quotients provide a structured approach to generating new Lipschitz ideals. Specifically, if a Lipschitz ideal $I_{\mathrm{Lip}}$ satisfies the linearization property within a given operator ideal $I$, then the left-hand quotient $A^{-1} \circ I_{\mathrm{Lip}}$ can be expressed as a Lipschitz composition ideal in the form $(A^{-1} \circ I)\circ I_{\mathrm{Lip}}$. Furthermore, we study Lipschitz mappings whose Lipschitz images are Grothendieck sets or Rosenthal sets, and demonstrate that both form Lipschitz ideals which can be constructed as Lipschitz left-hand quotients. 

 

 

Keywords

Summing operators

linearization property

Lipschitz operators

Subjects

• 47 Operator theory

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