Given a Banach space X and 1 ≤ p ≤ ∞, it is well known that the two Hardy spaces H p (T, X) (T the torus) and H p (D, X) (D the disk) have to be distinguished carefully. This motivates us to define and study two different types of Hardy spaces H p (X) and H + p (X) of Dirichlet series n a n n −s with coefficients in X. We characterize them in terms of summing operators as well as holomorphic functions in infinitely many variables, and prove that they coincide whenever X has the analytic Radon-Nikodým Property. Consequences are, among others, a vector-valued version of the Brother's Riesz Theorem in the infinite-dimensional torus, and an answer to the question when H 1 (X * ) is a dual space.
We introduce and study the notion of generating operators as those norm-one operators G : X −→ Y such that for every 0 < δ < 1, the set {x ∈ X : x 1, Gx > 1 − δ} generates the unit ball of X by closed convex hull. This class of operators includes isometric embeddings, spear operators (actually, operators with the alternative Daugavet property), and other examples like the natural inclusions of ℓ1 into c0 and of L∞[0, 1] into L1[0, 1]. We first present a characterization in terms of the adjoint operator, make a discussion on the behaviour of diagonal generating operators on c0-, ℓ1-, and ℓ∞-sums, and present examples in some classical Banach spaces. Even though rank-one generating operators always attain their norm, there are generating operators, even of rank-two, which do not attain their norm. We discuss when a Banach space can be the domain of a generating operator which does not attain its norm in terms of the behaviour of some spear sets of the dual space. Finally, we study when the set of all generating operators between two Banach spaces X and Y generates all non-expansive operators by closed convex hull. We show that this is the case when X = L1(µ) and Y has the Radon-Nikodým property with respect to µ. Therefore, when X = ℓ1(Γ), this is the case for every target space Y . Conversely, we also show that a real finite-dimensional space X satisfies that generating operators from X to Y generate all non-expansive operators by closed convex hull only in the case that X is an ℓ1-space.
The clinical and pathologic features of 6 cases of intestinal-type adenocarcinoma of the sinonasal region are presented. These cases were collected in a 17 year period (1972-1988) and account for less than 4% of malignancies of this region in our records for this period. All of the patients were men aged 48 to 82 years (mean, 54 years). Previous exposure to wood dust was reported in 1 case. Radiographic studies, especially computerized tomography, were of critical importance to delineate the extent of tumors. Nasal obstruction was the most common complaint. Duration of symptoms prior to diagnosis is available in 5 cases and ranged from 5 to 36 months (mean 18 months). Surgical treatment was performed in 4 patients (of palliative type in 2) followed by radiotherapy in 3. Histopathology revealed tubulo-papillary (5 cases) and mucinous (1 case) patterns. Follow-up is available in all patients (range 0 to 108 months), 50% of whom are still alive. In our series, only 1 patient has survived more than 5 years. Data pooled from the literature reveal that 53% of patients have experienced local recurrences following therapy, and 60% have died of their disease. Of these deaths, 80% occurred within 5 years of diagnosis.
Let f be a real-valued, degree-d Boolean function defined on the n-dimensional Boolean cube {±1} n , and f (x) = S⊂{1,...,d} f (S) k∈S x k its Fourier-Walsh expansion. The main result states that there is an absolute constant C > 0 such that the ℓ 2d/(d+1) -sum of the Fourier coefficients of f :It was recently proved that a similar result holds for complex-valued polynomials on the n-dimensional poly torus T n , but that in contrast to this a replacement of the n-dimensional torus T n by n-dimensional cube [−1, 1] n leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory.2010 Mathematics Subject Classification: Primary 06E30, 47A30. Secondary 81P45.
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