Some stability results in projective tensor products

TL;DR

This paper investigates whether certain properties of Fréchet spaces, specifically the property (y) and smallness of bounded sets, are preserved under projective tensor products, with implications for operator classes and complemented subspaces.

Contribution

It provides new results on the stability of properties (y) and smallness in projective tensor products of Fréchet spaces, extending understanding of their structural behavior.

Findings

Property (y) is preserved under projective tensor products under certain conditions.

Smallness of bounded sets can be stable in tensor products for specific classes of Fréchet spaces.

Results have implications for the stability of complemented subspaces and operator class equivalences.

Abstract

Let $(E,F)$ be a pair of Fr\'echet spaces. In this paper, we discuss whether a certain property $P$ enjoyed by both $E$ and $F$ is also satisfied by the complete tensor product $E \widehat{\otimes}_{\pi} F$. Specifically we focus on the two properties generalizing K\"othe spaces: the normability condition called the property $(y)$, and the property of smallness of bounded subsets of Fr\'echet spaces up to a complemented Banach subspace in connection with the problem of topologies of A. Grothendieck. We also consider the stability of some well known equivalences of operator classes with an application in the problem whether the sum of complemented subspaces is also complemented.