Universal Non-Completely-Continuous Operators

1995/04/13 by Girardi, Maria, Johnson, William B.
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.math/9504205

Abstract

A bounded linear operator between Banach spaces is called \it completely continuous if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators from L₁ into an arbitrary Banach space, namely, the operator from L₁ into ℓ_∞ defined by T₀ (f) =( ∫ r_n f dμ)_{n≥ 0} , where r_n is the n^th Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach space. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space.

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