Every separable, infinite-dimensional Banach space X has a biorthogonal sequence $z_n,z{*}_n$, with $spanz{*}_n$ norming on X and $\parallel z_n\parallel +\parallel z{*}_n\parallel$ bounded, so that, for every x in X and x* in X*, there exists a permutation π(n) of n so that $x\in \overline{conv}finitesubseriesof{\sum }_{n=1}^{\infty }z{*}_n\left(x\right)z_{n}andx{*}_n\left(x\right)={\sum }_{n=1}^{\infty }z{*}_{\pi \left(n\right)}\left(x\right)x*\left(z_{\pi \left(n\right)}\right)$.

It is proved that every operator from a weak*-closed subspace of ${\ell }_1$ into a space C(K) of continuous functions on a compact Hausdorff space K can be extended to an operator from ${\ell }_1$ to C(K).

A Banach space X is said to be an extremely non-complex space if the norm equality ∥Id +T 2∥ = 1+∥T 2∥ holds for every bounded linear operator T on X. We show that every extremely non-complex Banach space has positive numerical index, it does not have an unconditional basis and that the infimum of diameters of the slices of its unit ball is positive.

In some recent papers ([1],[2],[3],[4]) we have investigated some general spectral properties of a multiplier defined on a commutative semi-simple Banach algebra. In this paper we expose some aspects concerning the Fredholm theory of multipliers.

Suppose $H^2$ is the Hardy space of the unit disc in the complex plane, while $\Theta$ is an inner function. We give conditions for a sequence of normalized reproducing kernels in the model space $K_{\Theta }=H^2\ominus \Theta H^2$ to be asymptotically close to an orthonormal sequence. The completeness problem is also investigated.

It is known that Gabor expansions do not converge unconditionally in $L^p$ and that $L^p$ cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood-Paley and Gabor theory, we show that $L^p$ can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in $L^p$-norm.

A separable Banach space X contains ${\ell }_1$ isomorphically if and only if X has a bounded fundamental total $w{c}_0*$-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total $w{c}_0*$-biorthogonal system.

We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis in $L_p\left([0,1]\right)$, 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis ${}^d=\times ...\times$ in $L_p\left({[0,1]}^d\right)$,...

A Banach space is said to be $H{D}_n$ if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of $H{D}_n$ spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex $H{D}_n$, then dim $(ℒ\left(X\right)/S\left(X\right))\le n^2$, where S(X) denotes the space of strictly singular operators on X. It follows that if X is a real hereditarily indecomposable space, then ℒ(X)/S(X) is a division ring isomorphic either to ℝ, ℂ, or ℍ, the quaternionic division ring....

We introduce higher order spreading models associated to a Banach space X. Their definition is based on ℱ-sequences ${\left(x_s\right)}_{s\in ℱ}$ with ℱ a regular thin family and on plegma families. We show that the higher order spreading models of a Banach space X form an increasing transfinite hierarchy ${\left({ℳ}_{\xi }\left(X\right)\right)}_{\xi <\omega ₁}$. Each ${ℳ}_{\xi }\left(X\right)$ contains all spreading models generated by ℱ-sequences ${\left(x_s\right)}_{s\in ℱ}$ with order of ℱ equal to ξ. We also study the fundamental properties of this hierarchy.