A necessary and sufficient condition is given for a rearrangement invariant function space to contain a complemented isomorphic copy of l1(l2).

In this note we study some properties concerning certain copies of the classic Banach space $c_0$ in the Banach space $ℒ\left(X,Y\right)$ of all bounded linear operators between a normed space $X$ and a Banach space $Y$ equipped with the norm of the uniform convergence of operators.

It is proved that a separable Banach space X admits a representation $X=X_1+X_2$ as a sum (not necessarily direct) of two infinite-codimensional closed subspaces $X_1$ and $X_2$ if and only if it admits a representation $X=A_1\left(Y_1\right)+A_2\left(Y_2\right)$ as a sum (not necessarily direct) of two infinite-codimensional operator ranges. Suppose that a separable Banach space X admits a representation as above. Then it admits a representation $X=T_1\left(Z_1\right)+T_2\left(Z_2\right)$ such that neither of the operator ranges $T_1\left(Z_1\right)$, $T_2\left(Z_2\right)$ contains an infinite-dimensional closed subspace if and only...

Several conditions are given under which l1 embeds as a complemented subspace of a Banach space E if it embeds as a complemented subspace of an Orlicz space of E-valued functions. Previous results in Pisier (1978) and Bombal (1987) are extended in this way.

For a locally compact Hausdorff space K and a Banach space X let C₀(K, X) denote the space of all continuous functions f:K → X which vanish at infinity, equipped with the supremum norm. If X is the scalar field, we denote C₀(K, X) simply by C₀(K). We prove that for locally compact Hausdorff spaces K and L and for a Banach space X containing no copy of c₀, if there is an isomorphic embedding of C₀(K) into C₀(L,X), then either K is finite or |K| ≤ |L|. As a consequence, if there is an isomorphic embedding...

Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant $C_{NJ}\left(X\right)$, and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) and J(X*) are given as an answer to a problem of Gao and Lau [16]. Connections between $C_{NJ}\left(X\right)$ and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the $C_{NJ}\left(X\right)$-constant, which implies that a Banach space with $C_{NJ}\left(X\right)$-constant less than 5/4 has the fixed point property.

We obtain refinement of a result of Partington on Banach spaces containing isomorphic copies of l-∞. Motivated by this result, we prove that Banach spaces containing asymptotically isometric copies of l-∞ must contain isometric copies of l-∞.

We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ-spaces are of type ${ℒ}_{\infty }$ but not conversely. Moreover, ${ℒ}_{\infty }$-spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will show that...

Several properties of the class of minimal Orlicz function spaces LF are described. In particular, an explicitly defined class of non-trivial minimal functions is shown, which provides concrete examples of Orlicz spaces without complemented copies of F-spaces.

This paper introduces the class of Cohen p-nuclear m-linear operators between Banach spaces. A characterization in terms of Pietsch's domination theorem is proved. The interpretation in terms of factorization gives a factorization theorem similar to Kwapień's factorization theorem for dominated linear operators. Connections with the theory of absolutely summing m-linear operators are established. As a consequence of our results, we show that every Cohen p-nuclear (1 < p ≤ ∞ ) m-linear mapping...

Let 1 < p < ∞. Let X be a subspace of a space Z with a shrinking F.D.D. (Eₙ) which satisfies a block lower-p estimate. Then any bounded linear operator T from X which satisfies an upper-(C,p)-tree estimate factors through a subspace of ${\left(\sum Fₙ\right)}_{l_p}$, where (Fₙ) is a blocking of (Eₙ). In particular, we prove that an operator from $L_p$ (2 < p < ∞) satisfies an upper-(C,p)-tree estimate if and only if it factors through $l_p$. This gives an answer to a question of W. B. Johnson. We also prove that if X is...