Since 2011, the Spanish Functional Analysis Network, supported by over thirty five national resea... more Since 2011, the Spanish Functional Analysis Network, supported by over thirty five national research projects, organises the Escuela-Taller de Analisis Funcional yearly. More than two hundred undergraduate or master Mathematics students from twenty Spanish universities have taken part in this initiative. The 2019 edition of the school was dedicated to the memory of Bernardo Cascales, professor at the University of Murcia and promoter of the Escuela-Taller.
Journal of Mathematical Analysis and Applications, Sep 1, 2018
We study when the spaces of general Dirichlet series bounded on a half plane are Banach spaces, a... more We study when the spaces of general Dirichlet series bounded on a half plane are Banach spaces, and show that some of those classes are isometrically isomorphic between themselves. In a precise way, let {λ n } be a strictly increasing sequence of positive real numbers such that lim n→∞ λ n = ∞. We denote by H ∞ (λ n) the complex normed space of all Dirichlet series D(s) = n b n λ −s n , which are convergent and bounded on the half plane [Re s > 0], endowed with the norm D ∞ = sup Re s>0 |D(s)|. If (*) there exists q > 0 such that inf n (λ q n+1 − λ q n) > 0, then H ∞ (λ n) is a Banach space. Further, if there exists a strictly increasing sequence {r n } of positive numbers such that the sequence {log r n } is Q-linearly independent, μ n = r α for n = p α , and {λ n } is the increasing rearrangement of the sequence {μ n }, then H ∞ (λ n) is isometrically isomorphic to H ∞ (B c0). With this condition (*) we explain more explicitly the optimal cases of the difference among the abscissas σ c , σ b , σ u and σ a
The mathematical works of Manuel Valdivia (J. Horvath). Regularity properties of (LF)-spaces (D. ... more The mathematical works of Manuel Valdivia (J. Horvath). Regularity properties of (LF)-spaces (D. Vogt). Some applications of a decomposition method (A. Aytuna, T. Terzioglu). On the range of the Borel map for classes of non-quasianalytic functions (J. Bonet, R. Meise, B.A. Taylor). Biduality in Frechet and (LB)-spaces (K.D. Bierstedt, J. Bonet). Holomorphic mappings of bounded type on (DF)-spaces (P. Galindo, D. Garcia, M. Maestre). Linearization of holomorphic mappings of bounded type (J. Mujica). Spaces of holomorphic functions and germs on quotients (J.M. Ansemil, R.M. Aron, S. Ponte). Automatic continuity of intertwiners in topological vector spaces (K.B. Laursen). Barrelled function spaces (L. Drewnowski, M. Florencio, P.J. Paul). On distinguished Frechet spaces (J. Bonet, S. Dierolf). Prequojections and their duals (G. Metafune, V.B. Moscatelli). Problems from the Perez Carreras/Bonet book (S.A. Saxon). Interior properties and fixed points of certain discontinuous operators (W.R. Derrick, L. Nova G.). Functional analytic aspects of geometry. Linear extending of metrics and related problems (C. Bessaga). Lotosky-Schnabl operators on the unit interval and degenerate diffusion equations (F. Altomare). On weakly Lindelof Banach spaces (J. Orihuela). Distinguished subsets in vector sequence spaces (F. Bombal). Weak topologies on bounded sets of a Banach space. Associated function spaces (J.G. Llavona). Factorization of multilinear operators (J. Taskinen). Complex geodesics on convex domains (S. Dineen, R.M. Timoney). Continuity of tensor product operators between spaces of Bochner integrable functions (A. Defant, K. Floret). Compact convex sets in the two-dimensional complex linear space with the Yost property (E. Behrends). Some remarks on a limit class of approximation ideals (F. Cobos, T. Kuhn). Some factorization properties of composition operators (H. Jarchow). Eigenvalues of nuclear operators on Tsirelson spaces (A. Pietsch). Absolutely summing surjections from Sobolev spaces in the uniform norm (A. Pelczynski, M. Wojciechowski).
We study the Bishop-Phelps-Bollobás property for numerical radius restricted to the case of compa... more We study the Bishop-Phelps-Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that C0pLq spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of C0pLq spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of 1 have the BPBp-nu for compact operators. 1. Introduction, notation, and known results First we fix some notation in order to be able to describe our aims and results with precision. Given a Banach space X over the field K of real or complex numbers, we denote by X˚, B X , and S X , its topological dual, its closed unit ball, and its unit sphere, respectively. If Y is another Banach space, LpX, Y q represents the space of all bounded and linear operators from X to Y , and we denote by KpX, Y q the space of compact operators from X to Y. When Y " X, we shall simply write LpXq " LpX, Xq and KpXq " KpX, Xq. Given a locally compact Hausdorff topological space L, C 0 pLq is the Banach space of all scalar-valued continuous functions on L vanishing at infinity. Given an operator T P LpXq, its numerical radius is defined as νpT q :" sup t|x˚pT pxqq| : px, x˚q P ΠpXqu , where ΠpXq :" tpx, x˚q P S XˆSX˚: x˚pxq " 1u. It is immediate that νpT q ď }T } for every T P LpXq and that ν is a seminorm on LpXq. Very often, ν is actually a norm on LpXq equivalent to the usual operator norm. The numerical index of the space X measures this fact and it is given by npXq :" inftνpT q : T P LpXq, }T } " 1u " maxtk ě 0 : k}T } ď νpT q, @T P LpXqu.
The Hardy space H p of vector valued analytic functions in tube domains in C n and with values in... more The Hardy space H p of vector valued analytic functions in tube domains in C n and with values in Banach space are defined. Vector valued analytic functions in tube domains in C n with values in Hilbert space and which have vector valued tempered distributions as boundary value are proved to be in H p corresponding to Hilbert space if the boundary value is in L p with values in Hilbert space. A Poisson integral representation for such vector valued analytic
We characterize the uniform limits of Dirichlet polynomials on a right half plane. We extend the ... more We characterize the uniform limits of Dirichlet polynomials on a right half plane. We extend the approximation theorems of Runge, Mergelyan and Vitushkin to the Dirichlet setting with respect to the Euclidean distance and to the chordal one, as well. We also strengthen the notion of Universal Dirichlet series.
Our aim in this paper is to study weak compactness of composition operators between weighted spac... more Our aim in this paper is to study weak compactness of composition operators between weighted spaces of holomorphic functions on the unit ball of a Banach space.
Since 2011, the Spanish Functional Analysis Network, supported by over thirty five national resea... more Since 2011, the Spanish Functional Analysis Network, supported by over thirty five national research projects, organises the Escuela-Taller de Analisis Funcional yearly. More than two hundred undergraduate or master Mathematics students from twenty Spanish universities have taken part in this initiative. The 2019 edition of the school was dedicated to the memory of Bernardo Cascales, professor at the University of Murcia and promoter of the Escuela-Taller.
Journal of Mathematical Analysis and Applications, Sep 1, 2018
We study when the spaces of general Dirichlet series bounded on a half plane are Banach spaces, a... more We study when the spaces of general Dirichlet series bounded on a half plane are Banach spaces, and show that some of those classes are isometrically isomorphic between themselves. In a precise way, let {λ n } be a strictly increasing sequence of positive real numbers such that lim n→∞ λ n = ∞. We denote by H ∞ (λ n) the complex normed space of all Dirichlet series D(s) = n b n λ −s n , which are convergent and bounded on the half plane [Re s > 0], endowed with the norm D ∞ = sup Re s>0 |D(s)|. If (*) there exists q > 0 such that inf n (λ q n+1 − λ q n) > 0, then H ∞ (λ n) is a Banach space. Further, if there exists a strictly increasing sequence {r n } of positive numbers such that the sequence {log r n } is Q-linearly independent, μ n = r α for n = p α , and {λ n } is the increasing rearrangement of the sequence {μ n }, then H ∞ (λ n) is isometrically isomorphic to H ∞ (B c0). With this condition (*) we explain more explicitly the optimal cases of the difference among the abscissas σ c , σ b , σ u and σ a
The mathematical works of Manuel Valdivia (J. Horvath). Regularity properties of (LF)-spaces (D. ... more The mathematical works of Manuel Valdivia (J. Horvath). Regularity properties of (LF)-spaces (D. Vogt). Some applications of a decomposition method (A. Aytuna, T. Terzioglu). On the range of the Borel map for classes of non-quasianalytic functions (J. Bonet, R. Meise, B.A. Taylor). Biduality in Frechet and (LB)-spaces (K.D. Bierstedt, J. Bonet). Holomorphic mappings of bounded type on (DF)-spaces (P. Galindo, D. Garcia, M. Maestre). Linearization of holomorphic mappings of bounded type (J. Mujica). Spaces of holomorphic functions and germs on quotients (J.M. Ansemil, R.M. Aron, S. Ponte). Automatic continuity of intertwiners in topological vector spaces (K.B. Laursen). Barrelled function spaces (L. Drewnowski, M. Florencio, P.J. Paul). On distinguished Frechet spaces (J. Bonet, S. Dierolf). Prequojections and their duals (G. Metafune, V.B. Moscatelli). Problems from the Perez Carreras/Bonet book (S.A. Saxon). Interior properties and fixed points of certain discontinuous operators (W.R. Derrick, L. Nova G.). Functional analytic aspects of geometry. Linear extending of metrics and related problems (C. Bessaga). Lotosky-Schnabl operators on the unit interval and degenerate diffusion equations (F. Altomare). On weakly Lindelof Banach spaces (J. Orihuela). Distinguished subsets in vector sequence spaces (F. Bombal). Weak topologies on bounded sets of a Banach space. Associated function spaces (J.G. Llavona). Factorization of multilinear operators (J. Taskinen). Complex geodesics on convex domains (S. Dineen, R.M. Timoney). Continuity of tensor product operators between spaces of Bochner integrable functions (A. Defant, K. Floret). Compact convex sets in the two-dimensional complex linear space with the Yost property (E. Behrends). Some remarks on a limit class of approximation ideals (F. Cobos, T. Kuhn). Some factorization properties of composition operators (H. Jarchow). Eigenvalues of nuclear operators on Tsirelson spaces (A. Pietsch). Absolutely summing surjections from Sobolev spaces in the uniform norm (A. Pelczynski, M. Wojciechowski).
We study the Bishop-Phelps-Bollobás property for numerical radius restricted to the case of compa... more We study the Bishop-Phelps-Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that C0pLq spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of C0pLq spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of 1 have the BPBp-nu for compact operators. 1. Introduction, notation, and known results First we fix some notation in order to be able to describe our aims and results with precision. Given a Banach space X over the field K of real or complex numbers, we denote by X˚, B X , and S X , its topological dual, its closed unit ball, and its unit sphere, respectively. If Y is another Banach space, LpX, Y q represents the space of all bounded and linear operators from X to Y , and we denote by KpX, Y q the space of compact operators from X to Y. When Y " X, we shall simply write LpXq " LpX, Xq and KpXq " KpX, Xq. Given a locally compact Hausdorff topological space L, C 0 pLq is the Banach space of all scalar-valued continuous functions on L vanishing at infinity. Given an operator T P LpXq, its numerical radius is defined as νpT q :" sup t|x˚pT pxqq| : px, x˚q P ΠpXqu , where ΠpXq :" tpx, x˚q P S XˆSX˚: x˚pxq " 1u. It is immediate that νpT q ď }T } for every T P LpXq and that ν is a seminorm on LpXq. Very often, ν is actually a norm on LpXq equivalent to the usual operator norm. The numerical index of the space X measures this fact and it is given by npXq :" inftνpT q : T P LpXq, }T } " 1u " maxtk ě 0 : k}T } ď νpT q, @T P LpXqu.
The Hardy space H p of vector valued analytic functions in tube domains in C n and with values in... more The Hardy space H p of vector valued analytic functions in tube domains in C n and with values in Banach space are defined. Vector valued analytic functions in tube domains in C n with values in Hilbert space and which have vector valued tempered distributions as boundary value are proved to be in H p corresponding to Hilbert space if the boundary value is in L p with values in Hilbert space. A Poisson integral representation for such vector valued analytic
We characterize the uniform limits of Dirichlet polynomials on a right half plane. We extend the ... more We characterize the uniform limits of Dirichlet polynomials on a right half plane. We extend the approximation theorems of Runge, Mergelyan and Vitushkin to the Dirichlet setting with respect to the Euclidean distance and to the chordal one, as well. We also strengthen the notion of Universal Dirichlet series.
Our aim in this paper is to study weak compactness of composition operators between weighted spac... more Our aim in this paper is to study weak compactness of composition operators between weighted spaces of holomorphic functions on the unit ball of a Banach space.
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