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The Daugavet equation for polynomials

100%
Choi Y., García D., Maestre M., Martín M.
Studia Mathematica
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2007
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tom 178
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nr 1
63-84
EN
We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ||Id + P|| = 1 + ||P|| is satisfied for all weakly compact polynomials P: X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation $max_{|ω|=1} ||Id + ωP|| = 1 + ||P||$ for polynomials P: X → X. We show that this equation holds for every polynomial on the complex space X = C(K) (K arbitrary) with values in X. This result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for k-homogeneous polynomials.
2
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Integral holomorphic functions

100%
Dimant V., Galindo P., Maestre M., Zalduendo I.
Studia Mathematica
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2004
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tom 160
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nr 1
83-99
EN
We define the class of integral holomorphic functions over Banach spaces; these are functions admitting an integral representation akin to the Cauchy integral formula, and are related to integral polynomials. After studying various properties of these functions, Banach and Fréchet spaces of integral holomorphic functions are defined, and several aspects investigated: duality, Taylor series approximation, biduality and reflexivity.
3
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The Dirichlet-Bohr radius

88%
Carando D., Defant A., Garcí D., Maestre M., Sevilla-Peris P.
Acta Arithmetica
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2015
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tom 171
|
nr 1
23-37
EN
Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial $∑_{n ≤ x} a_n n^{-s}$ we have $∑_{n ≤ x} |a_n| r^{Ω(n)} ≤ sup_{t∈ ℝ} |∑_{n ≤ x} a_n n^{-it}|$. We prove that the asymptotically correct order of L(x) is $(log x)^{1/4} x^{-1/8}$. Following Bohr's vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa.
4
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Extension of multilinear mappings on Banach spaces

63%
Galindo P., Garciá D., Maestre M., Mujica J.
Studia Mathematica
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1994
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tom 108
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nr 1
55-76
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