### Browsing by Author "Gallardo Gutiérrez, Eva A."

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Publication A continuous model for quasinilpotent operators.(Springer, 2016-05-11) Gallardo Gutiérrez, Eva A.; Partington, Jonathan R.; Rodriguez, Daniel J.A classical result due to Foias and Pearcy establishes a discrete model for every quasinilpotent operator acting on a separable, infinite-dimensional complex Hilbert space HH . More precisely, given a quasinilpotent operator T on HH , there exists a compact quasinilpotent operator K in HH such that T is similar to a part of K⊕K⊕⋯⊕K⊕⋯K⊕K⊕⋯⊕K⊕⋯ acting on the direct sum of countably many copies of HH . We show that a continuous model for any quasinilpotent operator can be provided. The consequences of such a model will be discussed in the context of C0C0 -semigroups of quasinilpotent operators.Publication A generalization of the Aleksandrov operator and adjoints of weighted composition operators(Association des Annales de l'Institut Fourier, 2013) Gallardo Gutiérrez, Eva A.; Partington, Jonathan R.A generalization of the Aleksandrov operator is provided, in order to represent the adjoint of a weighted composition operator on H-2 by means of an integral with respect to a measure. In particular, we show the existence of a family of measures which represents the adjoint of a weighted composition operator under fairly mild assumptions, and we discuss not only uniqueness but also the generalization of Aleksandrov-Clark measures which corresponds to the unweighted case, that is, to the adjoint of composition operators.Publication A new class of operators and a description of adjoints of composition operators(Elsevier, 2006) Cowen, Carl C.; Gallardo Gutiérrez, Eva A.Starting with a general formula, precise but difficult to use, for the adjoint of a composition operator on a functional Hilbert space, we compute an explicit formula on the classical Hardy Hilbert space for the adjoint of a composition operator with rational symbol. To provide a foundation for this formula, we study an extension to the definitions of composition, weighted composition, and Toeplitz operators to include symbols that are multiple-valued functions. These definitions can be made on any Banach space of analytic functions on a plane domain, but in this work, our attention is focused on the basic properties needed for the application to operators on the standard Hardy and Bergman Hilbert spaces on the unit diskPublication A Paley–Wiener theorem for Bergman spaces with application to invariant subspaces(London Mathematical Society, 2007) Duren, Peter; Gallardo Gutiérrez, Eva A.; Montes Rodríguez, AlfonsoAn analogue of the Paley-Wiener theorem is developed for weighted Bergman spaces of analytic functions in the upper half-plane. The result is applied to show that the invariant subspaces of the shift operator on the standard Bergman space of the unit disk can be identified with those of a convolution Volterra operator on the space L 2 (ℝ + ,(1/t)dt).Publication Adjoints of linear fractional composition operators on the Dirichlet space(Springer, 2003) Gallardo Gutiérrez, Eva A.; Montes Rodríguez, AlfonsoThe adjoint of a linear fractional composition operator acting on the classical Dirichlet space is expressed as another linear fractional composition operator plus a two rank operator. The key point is that, in the Dirichlet space modulo constant functions, many linear fractional composition operators are similar to multiplication operators and, thus, normal. As a particular application, we can easily deduce the spectrum of each linear fractional composition operator acting on such spaces. Even the norm of each linear fractional composition operator is computed on the Dirichlet space modulo constant functions. It is also shown that all this work can be carried out in the Hardy space of the upper half plane.Publication An extension of a theorem of Domar on invariant subspaces(Univ. Szeged, Bolyai Institute, Aradi Vertanuk Tere, 2017) Gallardo Gutiérrez, Eva A.; Partington, J.R.; Rodriguez, D. J.A remarkable theorem of Domar asserts that the lattice of the invariant subspaces of the right shift semigroup {S-tau}(tau >= 0) in L-2 (R+, w(t)dt) consists of just the " standard invariant subspaces" whenever w is a positive continuous function in R+ such thatPublication Band-diagonal operators on Banach lattices: matrix dynamics and invariant subspaces(Elsevier Science, 2023) Gallardo Gutiérrez, Eva A.; González Doña, JavierWe address the existence of non-trivial closed invariant ideals for positive operators defined on Banach lattices whose order is induced by an unconditional basis. In particular, for band-diagonal positive operators such existence is characterized whenever their matrix representations meet a positiveness criteria. For more general classes of positive operators, sufficient conditions are derived proving, particularly, the sharpness of such results from the standpoint of view of the matrix representations. The whole approach is based on studying the behavior of the dynamics of infinite matrices and the localization of the non-zero entries. Finally, we generalize a theorem of Grivaux regarding the existence of non-trivial closed invariant subspaces for positive tridiagonal operators to a more general class of band-diagonal operators showing, in particular, that a large subclass of them have non-trivial closed invariant subspaces but lack non-trivial closed invariant ideals.Publication Boundedness, compactness and Schatten-class membership of weighted composition operators(Birkhauser Verlag, 2010) Gallardo Gutiérrez, Eva A.; Kumar, R.; Partington, Jonathan R.The boundedness and compactness of weighted composition operators on the Hardy space H(2) of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces A(alpha)(2)(D) are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.Publication Common hypercyclic vectors for families of operators(American Mathematical Society, 2008) Gallardo Gutiérrez, Eva A.; Partington, Jonathan R.We provide a criterion for the existence of a residual set of common hypercyclic vectors for an uncountable family of hypercyclic operators which is based on a previous one given by Costakis and Sambarino. As an application, we get common hypercyclic vectors for a particular family of hypercyclic scalar multiples of the adjoint of a multiplier in the Hardy space, generalizing recent results by Abakumov and Gordon and also Bayart. The criterion is applied to other specific families of operators.Publication Composition operators on Bergman spaces on Lavrentiev domains(Taylor & Francis, 2007) Gallardo Gutiérrez, Eva A.; González, María J.In this note, composition operators on Bergman spaces of a simply connected domain are studied characterizing boundedness and compactness of such operators whenever the domain is Lavrentiev.Publication Composition operators on hardy spaces on Lavrentiev domains(American Mathematical Society, 2008) Gallardo Gutiérrez, Eva A.; González, María J.; Nicolau, ArthurFor any simply connected domain., we prove that a Littlewood type inequality is necessary for boundedness of composition operators on H-p(Omega), 1 <= p < infinity, whenever the symbols are finitely-valent. Moreover, the corresponding "little-oh" condition is also necessary for the compactness. Nevertheless, it is shown that such an inequality is not sufficient for characterizing bounded composition operators even induced by univalent symbols. Furthermore, such inequality is no longer necessary if we drop the extra assumption on the symbol of being finitely-valent. In particular, this solves a question posed by Shapiro and Smith (2003). Finally, we show a striking link between the geometry of the underlying domain. and the symbol inducing the composition operator in H-p(Omega), and in this sense, we relate both facts characterizing bounded and compact composition operators whenever. is a Lavrentiev domain.Publication Consequences of universality among Toeplitz operators(Elsevier, 2015) Cowen, Carl C.; Gallardo Gutiérrez, Eva A.The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been one tool for studying the problem. The best known universal operators have been adjoints of analytic Toeplitz operators or unitarily equivalent to them. We present many examples of Toeplitz operators whose adjoints are universal operators and exhibit some of their common properties. Some ways in which the invariant subspaces of these universal operators interact with operators in their commutants are given. Special attention is given to the closed subalgebra, not always the zero algebra, of compact operators in their commutants. Finally, three questions connecting shift invariant subspaces and invariant subspaces of analytic Toeplitz operators are raised. Positive answers for both of the first two imply the existence of non-trivial invariant subspaces for every bounded operator on separable Hilbert spaces of dimension two or more.Publication Consequences of universality among Toeplitz operators(Elsevier, 2015-12) Cowen, Carl C.; Gallardo Gutiérrez, Eva A.The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been one tool for studying the problem. The best known universal operators have been adjoints of analytic Toeplitz operators or unitarily equivalent to them. We present many examples of Toeplitz operators whose adjoints are universal operators and exhibit some of their common properties. Some ways in which the invariant subspaces of these universal operators interact with operators in their commutants are given. Special attention is given to the closed subalgebra, not always the zero algebra, of compact operators in their commutants. Finally, three questions connecting shift invariant subspaces and invariant subspaces of analytic Toeplitz operators are raised. Positive answers for both of the first two imply the existence of non-trivial invariant subspaces for every bounded operator on separable Hilbert spaces of dimension two or more.Publication Cyclic behavior of linear fractional composition operators(Universidad de Extremadura, Departamento de Matemáticas, 2001) Gallardo Gutiérrez, Eva A.; Montes Rodríguez, AlfonsoPublication Cyclic Blaschke products for composition operators(Universidad Autónoma Madrid, 2009) Gallardo Gutiérrez, Eva A.; Gorkin, PamelaIn this work, cyclic Blaschke products for composition operators induced by disc automorphisms axe studied. In particular, we obtain interpolating Blaschke products that axe cyclic for nonelliptic automorphisms and we obtain a new characterization of Blaschke products that are not finite products of interpolating Blaschke products.Publication Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts(The Theta Foundation, 2009) Gallardo Gutiérrez, Eva A.; Partington, Jonathan R.; Segura, DoloresIt is shown that the invariant subspaces for the Bergman and Dirichlet shifts on the right half-plane correspond to the common invariant subspaces of the right shift operators on certain weighted Lebesgue spaces on the half-line. As a particular instance, the corresponding result for invariant subspaces of multipliers induced by weak-star generators of H(infinity)(D) on weighted Bergman spaces of the unit disc is deduced. Finally, cyclic vectors for the Bergman and Dirichlet shifts are also studied.Publication Distribution of primes and approximation on weighted Dirichlet spaces(2022) Gallardo Gutiérrez, Eva A.; Seco, DanielWe study zero-free regions of the Riemann zeta function ζ related to an approximation problem in the weighted Dirichlet space D−2 which is known to be equivalent to the Riemann Hypothesis since the work of B ́aez-Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces Dα when α ∈ (−3, −2) give conditions so that the half-plane {s ∈ C : R(s) > − α+12} is also zero-free for ζ. Moreover, we extend such results to a large family of weighted spaces of analytic functions lp α. As a particular instance, in the limit case p = 1 and α = −2, we provide a new proof of the Prime Number Theorem.Publication Exceptional sets and Hilbert–Schmidt composition operators(Elsevier, 2003) Gallardo Gutiérrez, Eva A.; González, María J.It is shown that an analytic map phi of the unit disk into itself inducing a Hilbert-Schmidt composition operator on the Dirichlet space has the property that the set E-phi = {e(i0)is an element ofpartial derivativeD : \phi(e(10))\ = 1 has zero logarithmic capacity. We also show that this is no longer true for compact composition operators on the Dirichlet space. Moreover, such a condition is not even satisfied by Hilbert-Schmidt composition operators on the Hardy space.Publication Finite rank perturbations of normal operators: spectral idempotents and decomposability(ELSEVIER, 2023-08-30) Gallardo Gutiérrez, Eva A.; González Doña, F. JavierWe prove that a large class of finite rank perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space are decomposable operators in the sense of Colojoară and Foiaş [1]. Consequently, every operator T in such a class has a rich spectral structure and plenty of non-trivial closed hyperinvariant subspaces which extends, in particular, previous theorems of Foiaş, Jung, Ko and Pearcy [5], [6], [7], Fang and J. Xia [3] and the authors [8], [9] on an open question posed by Pearcy in the seventies.Publication Generators of C0-Semigroups of weighted composition operators(2021-10-12) Gallardo Gutiérrez, Eva A.; Siskakis, Aristomenis G.; Yakubovich, DmitryWe prove that in a large class of Banach spaces of analytic functions inthe unit disc D an (unbounded) operator Af = G · f′ + g · f with G, g analytic in D generates a C0-semigroup of weighted composition operators if and only if it generates a C0-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces X of analytic functions in D contained in the Bloch space, we prove that no non-trivial holomorphic flow induces a C0-semigroup of weighted composition operators on X. This generalizes previous results in [6] and [1] regarding C0-semigroup of (unweighted) composition operators.

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