Schauder-Orlicz decompositions, $\ell_{\Phi}$-decompositions and pseudo-Daugavet property

Abstract

The concept of $\ell_{\Phi}$-decomposition, extending the concept of $\ell_{p}$-decomposition of a Banach space, is presented and basic properties of Schauder-Orlicz decompositions and $\ell_{\Phi}$-decompositions are studied. We show that Schauder-Orlicz decompositions are orthogonal in a sense of Grinblyum-James and Singer. Simple constructions of $\ell_{p}$-decompositions and Schauder-Orlicz decompositions in $L_p$ are presented. We prove that in the class of spaces possessing pseudo-Daugavet property, which includes classical $L_p$, $1\leq p\neq 2$, and $C$, Schauder-Orlicz decompositions with at least one finite dimensional subspace do not exist. It follows that Kato theorem on similarity for sequences of projections [1] cannot be extended to spaces from this class. Moreover we show that Banach spaces, possessing Schauder-Orlicz decompositions with at least one finite dimensional subspace, do not have pseudo-Daugavet property. Thus for Banach spaces $X$ possessing Schauder-Orlicz decompositions we obtain the following characterization of pseudo-Daugavet property: $X$ has pseudo-Daugavet property if and only if there is no Schauder-Orlicz decomposition in $X$ with at least one finite dimensional subspace if and only if there is no Schauder-Orlicz decomposition in $X$, which is an FDD.