Banach spaces without approximation properties of type p

TL;DR

This paper demonstrates that certain Banach spaces lack approximation properties of type p, resolving a question previously posed and showing the result was known since the 1980s through earlier research.

Contribution

It clarifies that the question about approximation properties of type p in Banach spaces has a negative answer, confirming prior findings from the 1980s.

Findings

Banach spaces without approximation properties of type p.

Negative answer to the posed question about factorization.

Prior results from the 1980s confirm the main conclusion.

Abstract

The main purpose of this note is to show that the question posed in the paper of Sinha D.P. and Karn A.K.("Compact operators which factor through subspaces of $l_p$ Math. Nachr. 281, 2008, 412-423; see the very end of that paper) has a negative answer, and that the answer was known, essentially, in 1985 after the papers "Approximation properties of order p and the existence of non-p-nuclear operators with p-nuclear second adjoints" (Math. Nachr. 109(1982), 125-134) and "Approximation of operators in Banach spaces" (Application of functional analysis in the approximation theory (KGU, Kalinin), 1985, 128-142) by Reinov O.I. have been appeared in 1982 and in 1985 respectively.