We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces ${\ell }_p\left(X\right)$, where X is a Banach space with a 1-unconditional basis and p ∈ (1,2) ∪ (2,∞). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of ${\ell }_p\left(X\right)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then...

In this survey we show that the separable quotient problem for Banach spaces is equivalent to several other problems for Banach space theory. We give also several partial solutions to the problem.

The present paper is a continuation of [23], from which we know that the theory of traces on the Marcinkiewicz operator ideal $\left(H\right):=T\in \left(H\right):su{p}_{1\le m<\infty }1/(logm+1){\sum }_{n=1}^{m}aₙ\left(T\right)<\infty$ can be reduced to the theory of shift-invariant functionals on the Banach sequence space $\left(\mathbb{N}₀\right):=c=\left({\gamma }_l\right):su{p}_{0\le k<\infty }1/(k+1){\sum }_{l=0}^k\left|{\gamma }_l\right|<\infty$. The final purpose of my studies, which will be finished in [24], is the following. Using the density character as a measure, I want to determine the size of some subspaces of the dual *(H). Of particular interest are the sets formed by the Dixmier traces and the Connes-Dixmier traces...

Let ${}_{\infty }\left(B_X\right)$ be the Banach space of all bounded and continuous functions on the closed unit ball $B_X$ of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let ${}_u\left(B_X\right)$ be the subspace of ${}_{\infty }\left(B_X\right)$ of those functions which are uniformly continuous on $B_X$. A subset $B\subset B_X$ is a boundary for ${}_{\infty }\left(B_X\right)$ if $\parallel f\parallel =su{p}_{x\in B}\left|f\left(x\right)\right|$ for every $f{\in }_{\infty }\left(B_X\right)$. We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for ${}_{\infty }\left(B_X\right)$. On the other hand, for X = , the Schreier space,...

2000 Mathematics Subject Classification: 45G15, 26A33, 32A55, 46E15.Schauder's fixed point theorem is used to establish an existence result for an infinite system of singular integral equations in the form: (1) xi(t) = ai(t)+ ∫t0 (t − s)− α (s, x1(s), x2(s), …) ds, where i = 1,2,…, α ∈ (0,1) and t ∈ I = [0,T]. The result obtained is applied to show the solvability of an infinite system of differential equation of fractional orders.

We present monotonicity theorems for index functions of N-fuctions, and obtain formulas for exact values of packing constants. In particular, we show that the Orlicz sequence space $l^{\left(N\right)}$ generated by the N-function N(v) = (1+|v|)ln(1+|v|) - |v| with Luxemburg norm has the Kottman constant $K\left(l^{\left(N\right)}\right)=N^{-1}\left(1\right)/N^{-1}(1/2)$, which answers M. M. Rao and Z. D. Ren’s [8] problem.