Projection from space of two-Lipschitz operators onto the space of Bilinear maps

2025/12/05 by Arindam Mandal, Mandal, Arindam
Computer Science · Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Optimization and Variational Analysis #Primary 46B28 #Secondary 46B10

paper · pdf · doi:10.48550/arxiv.2512.05640

openalex publication_date 2025/12/05 · openalex created_date 2025/12/09 · openalex updated_date 2026/07/28

Abstract

In this article, we establish the existence of a norm-one projection from the space of all two-Lipschitz operators onto the space of all bounded bilinear operators under certain conditions on the corresponding codomain spaces, using the method of invariant means. We also show that, when the codomain is an injective Banach space, the quotient of the two-Lipschitz operator space by the bounded bilinear space is isometrically isomorphic to a specific operator space, via vector-valued duality. We conclude by proving a necessary and sufficient condition for a two-Lipschitz operator to be a bilinear map. As an application of the theory developed here, we present an alternative proof that \bigslantL^∞(ℝ× ℝ)span\1\ is a dual space.

Citations

• Position of L(X, Y) in Lip₀(X, Y)
• Lip-Linear operators and their connection to Lipschitz tensor products
• Building ideals of two-Lipschitz operators between mertic and Banach spaces
• Invariant means on Abelian groups capture complementability of Banach\n spaces in their second duals
• Product spaces generated by bilinear maps and duality

Related

• A duality operators/Banach spaces
• Position of L(X, Y) in Lip₀(X, Y)
• Lipschitz geometry of operator spaces and Lipschitz-free operator spaces
• Lip-Linear operators and their connection to Lipschitz tensor products
• On some new metric characterisations of Hilbert spaces