We study the c₀-content of a seminormalized basic sequence (χₙ) in a Banach space, by the use of ordinal indices (taking values up to ω₁) that determine dichotomies at every ordinal stage, based on the Ramsey-type principle for every countable ordinal, obtained earlier by the author. We introduce two such indices, the c₀-index ${\xi }^{\left(\chi ₙ\right)}₀$ and the semibounded completeness index ${\xi }_b^{\left(\chi ₙ\right)}$, and we examine their relationship. The countable ordinal values that these indices can take are always of the form ${\omega }^{\zeta }$. These results...

Necessary and sufficient conditions are given for Orlicz sequence spaces equipped with the Orlicz norm to be uniformly rotund in a weakly compact set of directions, using only conditions on the generating function of the space.

We show how one can, in a unified way, calculate the Kottman and the packing constants of the Orlicz sequence space defined by an N-function, equipped with either the gauge or Orlicz norms. The values of these constants for a class of reflexive Orlicz sequence spaces are found, using a quantitative index of N-functions and some interpolation theorems. The exposition is essentially selfcontained.

Projections onto the spaces of all Toeplitz operators on the N-torus and the unit sphere are constructed. The constructions are also extended to generalized Toeplitz operators and applied to show hyperreflexivity results.

We present a simple proof of a theorem that yields as a corollary a result of Valdivia that sharpens an old result of Johnson and Rosenthal.

In this paper we give estimations of Istratescu measure of noncompactness I(X) of a set X C lp(E1,...,En) in terms of measures I(Xj) (j=1,...,n) of projections Xj of X on Ej. Also a converse problem of finding a set X for which the measure I(X) satisfies the estimations under consideration is considered.

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.