We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA), a generalization of the Weak Orthogonal Matching Pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain cond...

We prove that a homogeneous Banach space B on the unit circle T can be embedded as a closed subspace of a dual space ΞB contained in the space of bounded Borel measures on T in such a way that the map B → ΞB defines a bijective correspondence between the class of homogeneous Banach spaces on T and the class of prehomogeneous Banach spaces on T. We apply our results to show that the algebra of a...

We show that, if E is a Banach space with a shrinking basis satisfying a certain condition, then the Banach algebra l∞(K(l2⊕E)) is not amenable; in particular, this is true for E = l with p ∈ (1,∞). As a consequence, l∞(K(E)) is not amenable for any infinite-dimensional Lp-space. This, in turn, entails the non-amenability of B(lp(E)) for any Lp-space E, so that, in particular, B(lp) and B(Lp[0,...

We introduce the sequence space ℓpλ(B) of none absolute type which is a p-normed space and BK space in the cases 0<p<1 and 1≤p≤∞, respectively, and prove that ℓpλ(B) and ℓ p are linearly isomorphic for 0<p≤∞. Furthermore, we give some inclusion relations concerning the space ℓpλ(B) and we construct the basis for the space ℓpλ(B), where 1≤p<∞. Furthermore, we determine the alpha-, beta- and gamm...

We establish triviality of some holomorphic Banach vector bundles on the maximal ideal space M(H∞) of the Banach algebra H∞ of bounded holomorphic functions on the unit disk D ⊂ C with pointwise multiplication and supremum norm. We apply the result to the study of the Sz.-Nagy operator corona problem. 1. Formulation of Main Results 1.1. We continue our study started in [Br2] of analytic objects...

Let (Q.,l,ß) be a probability space, X a Banach space, and L (fi, X) the Banach space of Bochner integrable functions f: Í2 —> X . Let W = {/ € Ll(p, X) : for a.e. w in Í2, ||/(w)|| < 1} . In this paper we characterize the rwc (relatively weakly compact) subsets of L (ß, X). The main results are as follows: Theorem A. A subset H of W is rwc iff given any sequence (/„) in H there exists a sequen...

Let (Ω,Σ) be a measurable space and let X be a Banach space. Throughout this paper G will be a countably additive vector measure G : Σ → X. Consider the space L1(G) of (classes of real) G-integrable functions, following the definition of Bartle, Dunford and Schwartz [1] and Lewis [9]. The properties of this space have been studied by Kluvánek and Knowles [7], Okada [10] and Curbera [2]. In the ...

We construct a quasi-Banach space which cannot be given an equivalent plurisubharmonic quasi-norm, but such that it has a quotient by a onedimensional space which is a Banach space. We then use this example to construct a compact convex set in a quasi-Banach space which cannot be atfinely embedded into the space L0 of all measurable functions.

We prove that the multiplier algebra of the Drury-Arveson Hardy space H n on the unit ball in C n has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson in the unit disk. This result is obtained as a corollary of the Toeplitz Corona Theorem and a new Banach space result: the Besov-Sobolev space B p has the ”baby corona property” for all 0 ≤ σ < n p ...

We propose an abstract framework for the computation of the spectrum (A) of a linear operator A : D(A) X ! X on a Banach space X through a condition in a smaller Banach space X 1. If this space is nite dimensional this yields a characteristic equation for (A). The method is tested for delay, integro-diierential and population equations and is applicable to control and stability questions. The s...