Journal of Mathematical Analysis and Applications, 2021
Abstract We exhibit a class of functions f : R → R which are bounded, continuous on R ∖ Q , left ... more Abstract We exhibit a class of functions f : R → R which are bounded, continuous on R ∖ Q , left discontinuous on Q , right differentiable on Q , and upper left Dini differentiable on R ∖ Q . Other properties of these functions, such as jump sizes and local extrema, are also discussed. These functions are constructed using probabilistic methods. We also show that the families of functions satisfying similar properties contain large algebraic structures (obtaining lineability, algebrability and coneability).
We consider $A(\Omega)$, the Banach space of functions $f$ from $ \overline{\Omega}=\prod_{i \in ... more We consider $A(\Omega)$, the Banach space of functions $f$ from $ \overline{\Omega}=\prod_{i \in I} \overline{U_i}$ to $\mathbb{C}$ that are continuous with respect to the product topology and separately holomorphic, where $I$ is an arbitrary set and $U_i$ are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off $ \overline{U_i}$ are uniformly dense in $A(\Omega)$. This generalizes previous results where $U_i=\mathbb{D}$ is the open unit disc in $\mathbb{C}$ or $\overline{U_i}^c$ is connected.
We study the group invariant continuous polynomials on a Banach space X that separate a given set... more We study the group invariant continuous polynomials on a Banach space X that separate a given set K in X and a point z outside K. We show that if X is a real Banach space, G is a compact group of L(X), K is a G-invariant set in X, and z is a point outside K that can be separated from K by a continuous polynomial Q, then z can also be separated from K by a G-invariant continuous polynomial P. It turns out that this result does not hold when X is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that X has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group invariant polynomial from the closed unit ball.
We consider A(Ω), the Banach space of functions f from Ω¯¯¯¯=∏i∈IUi¯¯¯¯¯ to C that are continuous... more We consider A(Ω), the Banach space of functions f from Ω¯¯¯¯=∏i∈IUi¯¯¯¯¯ to C that are continuous with respect to the product topology and separately holomorphic, where I is an arbitrary set and Ui are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off Ui¯¯¯¯¯ are uniformly dense in A(Ω). This generalizes previous results where Ui=D is the open unit disc in C or Ui¯¯¯¯¯c is connected.
In this paper, we study geometric properties of the set of group invariant continuous linear func... more In this paper, we study geometric properties of the set of group invariant continuous linear functionals and operators between Banach spaces. In particular, we present group invariant versions of the Hahn-Banach separation theorem and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators, establishing group invariant versions of the properties α of Schachermayer and β of Lindenstrauss, and presenting relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain’s result, establishing that if X has the Radon-Nikodým property, then X has the G-Bishop-Phelps theorem for G-invariant operators whenever G is a compact group.
We present a sufficient condition to ensure the density of the set of rational functions with pre... more We present a sufficient condition to ensure the density of the set of rational functions with prescribed poles in the algebra A ∞ (Ω).
For compact sets K ⊂ C d , we introduce a subalgebra A D (K) of A(K), which allows us to obtain M... more For compact sets K ⊂ C d , we introduce a subalgebra A D (K) of A(K), which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.
In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$... more In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$ there is a function f holomorphic in the unit disc, having $\psi $ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions f of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.
We use approximation in measure to solve an asymptotic Dirichlet problem on arbitrary open sets a... more We use approximation in measure to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann hypothesis are suggested.
Journal of Mathematical Analysis and Applications, 2020
Abstract Let H ∞ ( B c 0 ) be the algebra of all bounded holomorphic functions on the open unit b... more Abstract Let H ∞ ( B c 0 ) be the algebra of all bounded holomorphic functions on the open unit ball of c 0 and M ( H ∞ ( B c 0 ) ) the spectrum of H ∞ ( B c 0 ) . We prove that for any point z in the closed unit ball of l ∞ there exists an analytic injection of the open ball B l ∞ into the fiber of z in M ( H ∞ ( B c 0 ) ) , which is an isometry from the Gleason metric of B l ∞ to the Gleason metric of M ( H ∞ ( B c 0 ) ) . We also show that, for some Banach spaces X, B l ∞ can be analytically injected into the fiber M z ( H ∞ ( B X ) ) for every point z ∈ B X .
We study the existence of algebras of hypercyclic vectors for weighted backward shifts on Fréchet... more We study the existence of algebras of hypercyclic vectors for weighted backward shifts on Fréchet sequence spaces that are algebras when endowed with coordinatewise multiplication or with the Cauchy product. As a particular case we obtain that the sets of hypercyclic vectors for Rolewicz's and MacLane's operators are algebrable.
We introduce and investigate the mth polarization constant of a Banach space X for the numerical ... more We introduce and investigate the mth polarization constant of a Banach space X for the numerical radius. We first show the difference between this constant and the original mth polarization constant associated with the norm by proving that the new constant is minimal if and only if X is strictly convex, and that there exists a Banach space which does not have an almost isometric copy of 2 1 , such that the second polarization constant for the numerical radius is maximal. We also give a negative answer for complex Banach spaces to the question of Choi and Kim (J Lond Math Soc (2) 54(1):135-147, 1996) whether m k=1 k m (m k) m! is the optimal upper bound for the mth polarization constant for arbitrary m ∈ N. Finally, we generalize the result of García et al.
Proceedings of the American Mathematical Society, 2020
We show that for any Banach space and any compact topological group G ⊂ L ( X ) G\subset L(X) suc... more We show that for any Banach space and any compact topological group G ⊂ L ( X ) G\subset L(X) such that the norm of X X is G G -invariant, the set of norm attaining G G -invariant functionals on X X is dense in the set of all G G -invariant functionals on X X , where a mapping f f is called G G -invariant if for every x ∈ X x\in X and every g ∈ G g\in G , f ( g ( x ) ) = f ( x ) f\big (g(x)\big )=f(x) . In contrast, we show also that the analog of Bollobás result does not hold in general. A version of Bollobás and James’ theorems is also presented.
We show that if U is an arbitrary open subset of a Riemann surface and ϕ an arbitrary continuous ... more We show that if U is an arbitrary open subset of a Riemann surface and ϕ an arbitrary continuous function on the boundary ∂U , then there exists a holomorphic functionφ on U such that, for every p ∈ ∂U ,φ(x) → ϕ(p), as x → p outside a set of density 0 at p relative to U. These "solutions to a boundary problem" are not unique. In fact they can be required to have interpolating properties and also to assume all complex values near every boundary point. Our result is new even for the unit disc. c
We show that the multiples of the backward shift operator on the spaces ℓ p , 1 ≤ p < ∞, or c 0 ,... more We show that the multiples of the backward shift operator on the spaces ℓ p , 1 ≤ p < ∞, or c 0 , when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of A-hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the Hadamard product, does not possess frequently hypercyclic algebras. On the other hand, we show that for any frequently hypercyclic operator T on any Banach space, F HC(T) is algebrable for a suitable product, and in some cases it is even strongly algebrable.
Given a proper holomorphic mapping g : Ω ⊆ C n −→ Ω ⊆ C n and an algebra of holomorphic functions... more Given a proper holomorphic mapping g : Ω ⊆ C n −→ Ω ⊆ C n and an algebra of holomorphic functions B (e.g. P(K) where K ⊂ Ω is a compact set, H(U), A(U) or H ∞ (U) where U is an open and bounded set with U ⊂ Ω), we study the subalgebra B g of all functions compatible with the equivalence relation defined by the proper mapping g. We provide alternative representations of these algebras and describe the fibers in their spectra. Among other examples we relate the algebras of functions that are invariant under permutations and the algebras of functions defined on the symmetrized polydisk. Keywords Symmetric holomorphic functions • Fibers • Algebras of holomorphic functions • Several complex variables • Symmetrized polydisk
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2016
We consider A(), the Banach algebra of all functions f from = i∈I U i to C that are continuous on... more We consider A(), the Banach algebra of all functions f from = i∈I U i to C that are continuous on with respect to the product topology and separately holomorphic in , where I is an arbitrary set and U i are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off U i are uniformly dense in A(). This generalizes previous results where U i = D is the open unit disc in C or U i c is connected.
Starting in 2007, the MFO publishes a preprint series which mainly contains research results rela... more Starting in 2007, the MFO publishes a preprint series which mainly contains research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs-Programme (RiP) and the Oberwolfach-Leibniz-Fellows (OWLF), but this can also include an Oberwolfach Lecture, for example. A preprint can have a size from 1-200 pages, and the MFO will publish it on its website as well as by hard copy. Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free hard copies (DIN A4, black and white copy) by surface mail. Of course, the full copy right is left to the authors. The MFO only needs the right to publish it on its website www.mfo.de as a documentation of the research work done at the MFO, which you are accepting by sending us your file. In case of interest, please send a pdf file of your preprint by email to or , respectively. The file should be sent to the MFO within 12 months after your stay as RiP or OWLF at the MFO. ri There are no requirements for the format of the preprint, except that the introduction should contain a short appreciation and that the paper size (respectively format) should be DIN A4, "letter" or "article". On the front page of the hard copies, which contains the logo of the MFO, title and authors, we shall add a running number (20XX-XX). We cordially invite the researchers within the RiP or OWLF programme to make use of this offer and would like to thank you in advance for your cooperation.
Journal of Mathematical Analysis and Applications, 2021
Abstract We exhibit a class of functions f : R → R which are bounded, continuous on R ∖ Q , left ... more Abstract We exhibit a class of functions f : R → R which are bounded, continuous on R ∖ Q , left discontinuous on Q , right differentiable on Q , and upper left Dini differentiable on R ∖ Q . Other properties of these functions, such as jump sizes and local extrema, are also discussed. These functions are constructed using probabilistic methods. We also show that the families of functions satisfying similar properties contain large algebraic structures (obtaining lineability, algebrability and coneability).
We consider $A(\Omega)$, the Banach space of functions $f$ from $ \overline{\Omega}=\prod_{i \in ... more We consider $A(\Omega)$, the Banach space of functions $f$ from $ \overline{\Omega}=\prod_{i \in I} \overline{U_i}$ to $\mathbb{C}$ that are continuous with respect to the product topology and separately holomorphic, where $I$ is an arbitrary set and $U_i$ are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off $ \overline{U_i}$ are uniformly dense in $A(\Omega)$. This generalizes previous results where $U_i=\mathbb{D}$ is the open unit disc in $\mathbb{C}$ or $\overline{U_i}^c$ is connected.
We study the group invariant continuous polynomials on a Banach space X that separate a given set... more We study the group invariant continuous polynomials on a Banach space X that separate a given set K in X and a point z outside K. We show that if X is a real Banach space, G is a compact group of L(X), K is a G-invariant set in X, and z is a point outside K that can be separated from K by a continuous polynomial Q, then z can also be separated from K by a G-invariant continuous polynomial P. It turns out that this result does not hold when X is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that X has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group invariant polynomial from the closed unit ball.
We consider A(Ω), the Banach space of functions f from Ω¯¯¯¯=∏i∈IUi¯¯¯¯¯ to C that are continuous... more We consider A(Ω), the Banach space of functions f from Ω¯¯¯¯=∏i∈IUi¯¯¯¯¯ to C that are continuous with respect to the product topology and separately holomorphic, where I is an arbitrary set and Ui are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off Ui¯¯¯¯¯ are uniformly dense in A(Ω). This generalizes previous results where Ui=D is the open unit disc in C or Ui¯¯¯¯¯c is connected.
In this paper, we study geometric properties of the set of group invariant continuous linear func... more In this paper, we study geometric properties of the set of group invariant continuous linear functionals and operators between Banach spaces. In particular, we present group invariant versions of the Hahn-Banach separation theorem and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators, establishing group invariant versions of the properties α of Schachermayer and β of Lindenstrauss, and presenting relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain’s result, establishing that if X has the Radon-Nikodým property, then X has the G-Bishop-Phelps theorem for G-invariant operators whenever G is a compact group.
We present a sufficient condition to ensure the density of the set of rational functions with pre... more We present a sufficient condition to ensure the density of the set of rational functions with prescribed poles in the algebra A ∞ (Ω).
For compact sets K ⊂ C d , we introduce a subalgebra A D (K) of A(K), which allows us to obtain M... more For compact sets K ⊂ C d , we introduce a subalgebra A D (K) of A(K), which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.
In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$... more In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$ there is a function f holomorphic in the unit disc, having $\psi $ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions f of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.
We use approximation in measure to solve an asymptotic Dirichlet problem on arbitrary open sets a... more We use approximation in measure to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann hypothesis are suggested.
Journal of Mathematical Analysis and Applications, 2020
Abstract Let H ∞ ( B c 0 ) be the algebra of all bounded holomorphic functions on the open unit b... more Abstract Let H ∞ ( B c 0 ) be the algebra of all bounded holomorphic functions on the open unit ball of c 0 and M ( H ∞ ( B c 0 ) ) the spectrum of H ∞ ( B c 0 ) . We prove that for any point z in the closed unit ball of l ∞ there exists an analytic injection of the open ball B l ∞ into the fiber of z in M ( H ∞ ( B c 0 ) ) , which is an isometry from the Gleason metric of B l ∞ to the Gleason metric of M ( H ∞ ( B c 0 ) ) . We also show that, for some Banach spaces X, B l ∞ can be analytically injected into the fiber M z ( H ∞ ( B X ) ) for every point z ∈ B X .
We study the existence of algebras of hypercyclic vectors for weighted backward shifts on Fréchet... more We study the existence of algebras of hypercyclic vectors for weighted backward shifts on Fréchet sequence spaces that are algebras when endowed with coordinatewise multiplication or with the Cauchy product. As a particular case we obtain that the sets of hypercyclic vectors for Rolewicz's and MacLane's operators are algebrable.
We introduce and investigate the mth polarization constant of a Banach space X for the numerical ... more We introduce and investigate the mth polarization constant of a Banach space X for the numerical radius. We first show the difference between this constant and the original mth polarization constant associated with the norm by proving that the new constant is minimal if and only if X is strictly convex, and that there exists a Banach space which does not have an almost isometric copy of 2 1 , such that the second polarization constant for the numerical radius is maximal. We also give a negative answer for complex Banach spaces to the question of Choi and Kim (J Lond Math Soc (2) 54(1):135-147, 1996) whether m k=1 k m (m k) m! is the optimal upper bound for the mth polarization constant for arbitrary m ∈ N. Finally, we generalize the result of García et al.
Proceedings of the American Mathematical Society, 2020
We show that for any Banach space and any compact topological group G ⊂ L ( X ) G\subset L(X) suc... more We show that for any Banach space and any compact topological group G ⊂ L ( X ) G\subset L(X) such that the norm of X X is G G -invariant, the set of norm attaining G G -invariant functionals on X X is dense in the set of all G G -invariant functionals on X X , where a mapping f f is called G G -invariant if for every x ∈ X x\in X and every g ∈ G g\in G , f ( g ( x ) ) = f ( x ) f\big (g(x)\big )=f(x) . In contrast, we show also that the analog of Bollobás result does not hold in general. A version of Bollobás and James’ theorems is also presented.
We show that if U is an arbitrary open subset of a Riemann surface and ϕ an arbitrary continuous ... more We show that if U is an arbitrary open subset of a Riemann surface and ϕ an arbitrary continuous function on the boundary ∂U , then there exists a holomorphic functionφ on U such that, for every p ∈ ∂U ,φ(x) → ϕ(p), as x → p outside a set of density 0 at p relative to U. These "solutions to a boundary problem" are not unique. In fact they can be required to have interpolating properties and also to assume all complex values near every boundary point. Our result is new even for the unit disc. c
We show that the multiples of the backward shift operator on the spaces ℓ p , 1 ≤ p < ∞, or c 0 ,... more We show that the multiples of the backward shift operator on the spaces ℓ p , 1 ≤ p < ∞, or c 0 , when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of A-hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the Hadamard product, does not possess frequently hypercyclic algebras. On the other hand, we show that for any frequently hypercyclic operator T on any Banach space, F HC(T) is algebrable for a suitable product, and in some cases it is even strongly algebrable.
Given a proper holomorphic mapping g : Ω ⊆ C n −→ Ω ⊆ C n and an algebra of holomorphic functions... more Given a proper holomorphic mapping g : Ω ⊆ C n −→ Ω ⊆ C n and an algebra of holomorphic functions B (e.g. P(K) where K ⊂ Ω is a compact set, H(U), A(U) or H ∞ (U) where U is an open and bounded set with U ⊂ Ω), we study the subalgebra B g of all functions compatible with the equivalence relation defined by the proper mapping g. We provide alternative representations of these algebras and describe the fibers in their spectra. Among other examples we relate the algebras of functions that are invariant under permutations and the algebras of functions defined on the symmetrized polydisk. Keywords Symmetric holomorphic functions • Fibers • Algebras of holomorphic functions • Several complex variables • Symmetrized polydisk
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2016
We consider A(), the Banach algebra of all functions f from = i∈I U i to C that are continuous on... more We consider A(), the Banach algebra of all functions f from = i∈I U i to C that are continuous on with respect to the product topology and separately holomorphic in , where I is an arbitrary set and U i are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off U i are uniformly dense in A(). This generalizes previous results where U i = D is the open unit disc in C or U i c is connected.
Starting in 2007, the MFO publishes a preprint series which mainly contains research results rela... more Starting in 2007, the MFO publishes a preprint series which mainly contains research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs-Programme (RiP) and the Oberwolfach-Leibniz-Fellows (OWLF), but this can also include an Oberwolfach Lecture, for example. A preprint can have a size from 1-200 pages, and the MFO will publish it on its website as well as by hard copy. Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free hard copies (DIN A4, black and white copy) by surface mail. Of course, the full copy right is left to the authors. The MFO only needs the right to publish it on its website www.mfo.de as a documentation of the research work done at the MFO, which you are accepting by sending us your file. In case of interest, please send a pdf file of your preprint by email to or , respectively. The file should be sent to the MFO within 12 months after your stay as RiP or OWLF at the MFO. ri There are no requirements for the format of the preprint, except that the introduction should contain a short appreciation and that the paper size (respectively format) should be DIN A4, "letter" or "article". On the front page of the hard copies, which contains the logo of the MFO, title and authors, we shall add a running number (20XX-XX). We cordially invite the researchers within the RiP or OWLF programme to make use of this offer and would like to thank you in advance for your cooperation.
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Papers by Javier Falcó