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The objective of this work is to introduce ourselves to the theory of absolutely summing operators, giving an overview of them, based on the first chapter of the book with the same name to this work found in the bibliography. To do this, we will see some important results of absolute and unconditional convergence, establishing the relationship between them. We will also define other types of convergence, such as sign convergence or disordered convergence, and we will establish a list of equivalences that will be very useful. At the end of the second chapter we will and the Grothendieck’s Theorem, which will establish a result that will be the heart of this work. Finally, in the last chapter we will study the well-known Schatten-von Neumann classes, where we will pay special attention to the Hilbert-Schmidt classes and the nuclear operators or the trace class operators, seeing the relationship between them and the p-summing operators., El objetivo de este trabajo es introducirnos en la teoría de los operadores absolutamente sumantes, dando una visión general de los mismos, basándonos en el primer capítulo del libro homónimo a este trabajo que se encuentra en la bibliografía. Para ello, veremos algunos resultados importantes de convergencia absoluta e incondicional, estableciendo la relación entre ellas. También definiremos otros tipos de convergencia, como la convergencia por signos o la convergencia desordenada, y estableceremos una lista de equivalencias que resultarán de gran utilidad. Al final del segundo capítulo encontraremos el teorema de Grothendieck, que establecerá un resultando que será el corazón de este trabajo. Para analizar, en el último capítulo estudiaremos las conocidas clases de Schattenvon Neumann, donde prestaremos especial atención a las clases de Hilbert-Schmidt y a los operadores nucleares o de la clase de la traza, viendo la relación entre ellos y los operdores p-sumantes.

A main problem of Operator Theory consists of studying which operators have an invariant subspace, or what properties they must have to possess it. In this project we will work on separable complex Hilbert spaces and we will begin by defining and viewing properties of the classical topologies that we will use. Later, we will continue studying compact operators and proving the existence of invariant subspace. Next, we will do the same with normal operators, relying on the spectral theorem. Finally, we will define what is known as an essentially normal operator, and we will study a fairly recent result on the existence of these invariant subspaces under certain hypotheses., Un problema principal de la Teoría de Operadores consiste en estudiar qué operadores tienen subespacio invariante, o que propiedades deben tener para poseerlo. En este proyecto trabajaremos sobre espacios de Hilbert separables complejos y comenzaremos definiendo y viendo propiedades de las topologías clásicas que usaremos. Después, continuaremos estudiando los operadores compactos y probando la existencia de subespacio invariante. A continuación, haremos lo mismo con los operadores normales, apoyándonos en el teorema espectral. Finalmente, definiremos lo que se conoce como operador esencialmente normal, y estudiaremos un resultado bastante reciente sobre la existencia de dichos subespacios invariantes bajo ciertas hipótesis.

A main topic in matrix analysis is the computation of the norms of Schur multipliers in the algebra of square matrices. The object of this work is to survey classical results of Schur about the multiplier norm of any matrix, for example, the Schur inequality, that it is very useful for delimit the Schur multipliers norm. Also, we study the Schur multiplier norm of a positive definite matrix, and furthermore, some properties of the determinants of this matrix product. Then we apply them to the study of those matrices that have extremal multiplier norms, where unitary matrices stand out. Finally, we make use of the results to estimate the norm of the triangular truncation operator in the algebra of square matrices., Un tema principal en el análisis matricial es el cálculo de las normas de multiplicadores de Schur en el álgebra de las matrices cuadradas. El objetivo de este trabajo es examinar los resultados clásicos de Schur sobre la norma de multiplicador de cualquier matriz, por ejemplo, la desigualdad de Schur, que es muy útil para acotar la norma de multiplicadores. También estudiaremos la norma de multiplicador de Schur de una matriz definida positiva, además de algunas propiedades de los determinantes de este producto matricial. Luego, aplicaremos los resultados al estudio de las matrices que tienen norma de multiplicador extremal, donde sobresalen las matrices unitarias. Por último, haremos uso de los resultados para estimar la norma del operador de truncanción triangular en el álgebra de las matrices cuadradas.

The aim of this work is to study in depth the alternating algorithm proposed by John von Neumann in 1933. This algorithm deals with the problem of finding the ortogonal projection of a point in a Hilbert space onto the intersection of two closed subspaces, in terms of the alternating orthogonal projections onto the respective subspaces. We will study the convergence of the method and its rate of convergence, as well as the generalization of such algorithm to more than two subspaces or to the case when the subspaces are replaced by convex subsets. Finally, we discuss some applications where the method plays an important role, like the approximate solution of systems of linear equations or the Dirichlet problem., El objetivo de este trabajo es estudiar en profundidad el algoritmo alternante propuesto por John von Neumann en 1933. Este algoritmo trata el problema de encontrar la proyección ortogonal de un punto en un espacio de Hilbert sobre la intersección de dos subespacios cerrados, en términos de la proyecciones ortogonales alternadas sobre los respectivos subespacios. Además estudiamos la convergencia del método y su ritmo de convergencia, así como la generalización de dicho algoritmo a más de dos subespacios, o el caso cuando los subespacios se reemplazan por subconjuntos convexos. Por ´ultimo, discutimos algunas aplicaciones donde el método desempeña un papel importante, como la aproximación de sistemas de ecuaciones lineales o el problema de Dirichlet.

Background: Tuberculosis is a major public health problem and most cases are concentrated in vulnerable populations. The objective was to describe the incidence rates trend in native and foreign population (2009-2018) in Madrid Region. Methods: Retrospective analysis of cases from the Tuberculosis Regional Registry of cases of Madrid Region 2009-2018. Annual incidence rates were calculated by country of birth (Spain, other), sex and age group (<15, 15-34, 35-44, 45-64,> 64), using the annual January 1st continuous register population. The infection rate trend and the annual percentage change (APC) were calculated, along with the best jointpoint adjustment using Jointpoint regression. Results: 7,696 cases were analyzed, 48.2% were foreign- born individuals. Average age in native population was 50 years old (SD: 23.96) and 35 (DS: 36.64) in foreignborn individuals (p<0.001). The overall incidence rate decreased from 17.30 in 2009 to 9.00 per 100,000 in 2018 and was higher in men. Pulmonary tuberculosis reduced from 11.90 to 6.55. Among native population, the incidence of TB fell from 10.29 to 5.24 with an APC of -7.3% (95%IC: -8.9; -5.7) (p<0.05), no jointpoint was identified. Among foreign-born individuals the incidence of tuberculosis declined from 46.54 to 25.49, a joint point was identified in 2013, observing an incidence decrease for the period 2009- 2013 and APC of -13.8% (IC95%: -17.5; -10.0). Conclusions: The global incidence rate in this period has decreased by approximately 7% per year. However, this reduction occurred mainly in native population. In foreignborn individuals the incidence decreased by approximately 14% during the 2009-2013 period, after this period there have been no significant incidence changes., Fundamentos: La tuberculosis (TB) continúa siendo un problema importante de salud pública, debido a que la mayoría de los casos se concentran en población vulnerable. El objetivo de este trabajo fue describir la tendencia de las tasas de incidencia en población autóctona y extranjera (2009-2018) en la Comunidad de Madrid (CM). Métodos: Se realizó un análisis retrospectivo de casos del Registro Regional de casos de Tuberculosis de la CM en el período 2009-2018. Se calcularon tasas de incidencia anual por cada 100.000 habitantes, por país de nacimiento (España, fuera de España), sexo y grupo de edad (<15, 15- 34, 35-44, 45-64,>64), utilizando las poblaciones de padrón continuo a 1 de enero de cada año. Se calculó la tendencia de las tasas de incidencia y el porcentaje anual de cambio (APC), así como el mejor ajuste del punto de inflexión utilizando la regresión de Jointpoint. Resultados: Se analizaron 7.696 casos, siendo el 48,2% en personas nacidas fuera de España. La edad media en población autóctona fue de 50 años (DS: 23,96) y 35 (DS: 36,64) en inmigrante (p<0,001). La tasa de incidencia global pasó de 17,30 por cada 100.000 habitantes en 2009 a 9 en 2018, siendo superior en hombres. La incidencia de tuberculosis pulmonar pasó de 11,90 a 6,55. En población autóctona, la incidencia de TB pasó de 10,29 a 5,24, con un APC de -7,3% (IC95%: -8,9; -5,7; p<0,05), y no se identificó ningún punto de inflexión. En población extranjera la incidencia de tuberculosis pasó de 46,54 a 25,49, identificándose un punto de inflexión en 2013, con una disminución más acusada de la incidencia para el periodo 2009-2013 debido a un APC de -13,8% (IC95%: -17,5; -10,0). Conclusiones: La tasa de incidencia global en este periodo disminuye cerca de un 7% anual; sin embargo, esta disminución de la incidencia se produce fundamentalmente en población autóctona. En población extranjera la incidencia desciende cerca de un 14% durante el periodo 2009- 2013. Tras este periodo no hay cambios signifi

The Invariant Subspace Problem is one of the most studied problems on Operator Theory in the last decades. In fact, it is still open in the Hilbert space setting. The purpose of this work is to present the most classic results concerning this problem and to study the approach based on universal operators. The text is organized as follows: In the first chapter we introduce the theory Banach algebras, focusing on spectral theory and Gelfand transform, two tools that will be fundamental in the development of the text. In the second chapter we provide a classical view of the invariant subspace problem in Hilbert spaces. We show two of the most important results on the existence of hyperinvariant subspaces: Lomonosov theorem and spectral theorem for normal operators. In the third chapter we study the tools to calculate invariant subspaces lattices for some classical operators, emphasizing on the need to use models to characterize these lattices. In chapter four we introduce the universal operators and we prove that the characterization of the invariant subspaces lattice of the hyperbolic automorphism composition operator in H2 would solve the invariant subspace problem. Finally, in chapter five we present the closest result to date: the characterization of the lattice of the parabolic nonautomorphism composition operator in H2., El Problema del Subespacio Invariante es uno de los problemas más estudiados en Teoría de Operadores en las últimas décadas. De hecho, sigue abierto para operadores definidos en espacios de Hilbert. El objetivo de este trabajo es presentar los resultados más clásicos en este problema y estudiar el enfoque basado en los operadores universales. El trabajo está organizado como sigue: En el primer capítulo, introducimos la teoría de álgebras de Banach, haciendo hincapié en la teoría espectral y en la transformada de Gelfand, dos herramientas que serán dos elementos fundamentales en el desarrollo del texto. En el capítulo segundo proporcionamos una visión clásica del problema del subespacio invariante en espacios de Hilbert. Demostramos dos de los resultados más importantes de existencia de subespacios hıperinvariantes: el teorema de Lomonosov y el teorema espectral para operadores normales. En el tercer capítulo, estudiamos las herramientas para calcular los retículos de subespacios invariantes de algunos operadores clásicos, enfatizando la necesidad del uso de modelos para caracterizar dichos retículos. En el capítulo cuarto introducimos los operadores universales y probamos que la caracterización del retículo de subespacios invariantes del operador de composición hiperbólico automorfismo en H2 resolvería el problema del subespacio invariante. Finalmente, en el quinto capítulo presentamos el resultado más cercano hasta la fecha: la caracterización del retículo del operador de composición parabólico no automorfismo en H2.

The main goal of this work is to characterize the membership of composition operators on the Hardy space H2 of the unit disk to the Schatten classes. Firstly, we will study the basic results for compact operators between Banach and Hilbert spaces, and based on the spectral theorem for compact operators we will introduce the operators ideal known as the Schatten classes and we will prove its more important properties. Secondly, we will introduce the Hardy space H2, focusing on composition operators defined on this space. We will prove Littlewood’s theorem, that shows that these operators are continuous, and we will study some basic results about compactness. Moreover, using the pull-back measure we will see that our main problem is equivalent to characterize the membership to Schatten classes of the embeddings from H2 to L2(µ), where µ is a Borel finite measure defined on the unit disc. Finally, we will study the continuity and compactness properties of these embeddings and we will present an alternative, new proof for Luecking’s theorem, that will solve our problem in the general framework and will give us the desired characterization of membership for composition operators to Schatten classes., El objetivo principal de este Trabajo de Fin de Grado es caracterizar la pertenencia a las clases de Schatten de los operadores de composición en el espacio de Hardy H2. Para ello, introduciremos y estudiaremos las propiedades básicas de los operadores compactos entre espacios de Banach y de Hilbert, y gracias al teorema espectral para operadores compactos entre espacios de Hilbert presentaremos los ideales de operadores conocidos como clases de Schatten y demostraremos sus características más importantes. A continuación, introduciremos el espacio de Hardy H2, centrándonos en los operadores de composición definidos sobre este espacio. Probaremos su continuidad gracias al teorema de Littlewood y estudiaremos sus propiedades más básicas de compacidad. Además, gracias al concepto de medida imagen, probaremos que nuestro problema inicial es equivalente a caracterizar la pertenencia a las clases de Schatten de las inyecciones de H2 en L2(µ), donde µ es una medida boreliana finita en el disco. Finalmente, estudiaremos las propiedades de continuidad y de compacidad de estas inyecciones mediante el concepto de medida de Carleson y presentaremos una prueba novedosa del teorema de Luecking, que resolverá nuestro problema en el marco general y nos dará la caracterización buscada de pertenencia a las clases de Schatten para los operadores de composición.

A famous inequality by von Neumann states that if T is a contraction on a Hilbert space and p is a polynomial, then kp(T)k ≤ sup{|p(z)| : z ∈ C, |z| ≤ 1}. As time went on, this inequality has given rise to a large variety of results estimulating this question. The natural way to generalize this inequality concerns contractions T1, . . . , Tn that commute on a common Hilbert space. Is it true that, for any polynomial p(z1, . . . , zn) in n variables, kp(T1, . . . , Tn)k ≤ sup{|p(z1, . . . , zn)| : zi ∈ C, |zi | ≤ 1, i = 1, . . . , n}? The answer is partial. The major steps in answering this question are due to T. Ando and N. Varopoulos, as much in positive and negative cases, respectively. The aim of this work is to present an elegant proof using dilation theory, whose main forefather is Sz. Nagy, of the original von Neumann’s inequality, as well as describing its generalization for two commuting contractions, and some counterexamples on a finite-dimensional Hilbert space, emphasizing the M. J. Crabb, A. M. Davie and J. A. Holbrook counterexamples.

The main goal of this work is to characterize the membership of composition operators on the Hardy space H2 of the unit disk to the Schatten classes. Firstly, we will study the basic results for compact operators between Banach and Hilbert spaces, and based on the spectral theorem for compact operators we will introduce the operators ideal known as the Schatten classes and we will prove its more important properties. Secondly, we will introduce the Hardy space H2, focusing on composition operators defined on this space. We will prove Littlewood’s theorem, that shows that these operators are continuous, and we will study some basic results about compactness. Moreover, using the pull-back measure we will see that our main problem is equivalent to characterize the membership to Schatten classes of the embeddings from H2 to L2(µ), where µ is a Borel finite measure defined on the unit disc. Finally, we will study the continuity and compactness properties of these embeddings and we will present an alternative, new proof for Luecking’s theorem, that will solve our problem in the general framework and will give us the desired characterization of membership for composition operators to Schatten classes., El objetivo principal de este Trabajo de Fin de Grado es caracterizar la pertenencia a las clases de Schatten de los operadores de composición en el espacio de Hardy H2. Para ello, introduciremos y estudiaremos las propiedades básicas de los operadores compactos entre espacios de Banach y de Hilbert, y gracias al teorema espectral para operadores compactos entre espacios de Hilbert presentaremos los ideales de operadores conocidos como clases de Schatten y demostraremos sus características más importantes. A continuación, introduciremos el espacio de Hardy H2, centrándonos en los operadores de composición definidos sobre este espacio. Probaremos su continuidad gracias al teorema de Littlewood y estudiaremos sus propiedades más básicas de compacidad. Además, gracias al concepto de medida imagen, probaremos que nuestro problema inicial es equivalente a caracterizar la pertenencia a las clases de Schatten de las inyecciones de H2 en L2(µ), donde µ es una medida boreliana finita en el disco. Finalmente, estudiaremos las propiedades de continuidad y de compacidad de estas inyecciones mediante el concepto de medida de Carleson y presentaremos una prueba novedosa del teorema de Luecking, que resolverá nuestro problema en el marco general y nos dará la caracterización buscada de pertenencia a las clases de Schatten para los operadores de composición.

Halmos says that one of the most recalcitrant unsolved problems in operator theory is the invariant subspace problem. The question has an easy formulation: Does every bounded linear operator on an infinite dimensional, separable complex Hilbert space have a nontrivial invariant subspace? The aim of this work is to present an introduction to the theory of invariant subspaces, developing some basic notions from functional analysis, the elementary theory of Banach algebras and the Riesz-Fredholm theory about the spectrum of a compact operator, in order to finally provide a proof of Lomonosov’s theorem., Halmos dice que uno de los problemas más recalcitrantes en teoría de operadores es el problema del subespacio invariante. La cuestión posee un sencillo enunciado: ¿Tiene cualquier operador lineal y acotado en un espacio de Hilbert separable, complejo e infinito-dimensional un subespacio invariante no trivial? El objetivo de este trabajo es presentar una introducción a la teoría de los subespacios invariantes, desarrollando algunas nociones básicas del análisis funcional, la teoría elemental de las álgebras de Banach y la teoría de Riesz-Fredholm acerca del espectro de un operador compacto, para poder ofrecer finalmente una demostración del teorema de Lomonosov.

It is shown that if the Deddens algebra DT associated with a quasinilpotent operator T on a complex Banach space is closed and localizing then T has a nontrivial closed hyperinvariant subspace.

It is shown that if the Deddens algebra DT associated with a quasinilpotent operator T on a complex Banach space is closed and localizing then T has a nontrivial closed hyperinvariant subspace.

It is shown that if the Deddens algebra DT associated with a quasinilpotent operator T on a complex Banach space is closed and localizing then T has a nontrivial closed hyperinvariant subspace.

An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernández-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator.

An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernández-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator.

An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernández-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator.

An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernández-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator.

An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernández-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator.

We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.

Let µ be a finite positive Borel measure with compact support K ⊆ C, and regard L∞(µ) as an algebra of multiplication operators on the Hilbert space L2(µ). Then consider the subalgebra A(K) of all continuous functions on K that are analytic on the interior of K, and the subalgebra R(K) defined as the uniform closure of the rational functions with poles outside K. Froelich and Marsalli showed that if the restriction of the measure µ to the boundary of K is discrete then the unit ball of A(K) is strongly precompact, and that if the unit ball of R(K) is strongly precompact then the restriction of the measure µ to the boundary of each component of C\K is discrete. The aim of this paper is to provide three examples that go to clarify the results of Froelich and Marsalli; in particular, it is shown that the converses to both statements are false.

We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.

Let µ be a finite positive Borel measure with compact support K ⊆ C, and regard L∞(µ) as an algebra of multiplication operators on the Hilbert space L2(µ). Then consider the subalgebra A(K) of all continuous functions on K that are analytic on the interior of K, and the subalgebra R(K) defined as the uniform closure of the rational functions with poles outside K. Froelich and Marsalli showed that if the restriction of the measure µ to the boundary of K is discrete then the unit ball of A(K) is strongly precompact, and that if the unit ball of R(K) is strongly precompact then the restriction of the measure µ to the boundary of each component of C\K is discrete. The aim of this paper is to provide three examples that go to clarify the results of Froelich and Marsalli; in particular, it is shown that the converses to both statements are false.

We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.

We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.

Let µ be a finite positive Borel measure with compact support K ⊆ C, and regard L∞(µ) as an algebra of multiplication operators on the Hilbert space L2(µ). Then consider the subalgebra A(K) of all continuous functions on K that are analytic on the interior of K, and the subalgebra R(K) defined as the uniform closure of the rational functions with poles outside K. Froelich and Marsalli showed that if the restriction of the measure µ to the boundary of K is discrete then the unit ball of A(K) is strongly precompact, and that if the unit ball of R(K) is strongly precompact then the restriction of the measure µ to the boundary of each component of C\K is discrete. The aim of this paper is to provide three examples that go to clarify the results of Froelich and Marsalli; in particular, it is shown that the converses to both statements are false.

We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.

Let µ be a finite positive Borel measure with compact support K ⊆ C, and regard L∞(µ) as an algebra of multiplication operators on the Hilbert space L2(µ). Then consider the subalgebra A(K) of all continuous functions on K that are analytic on the interior of K, and the subalgebra R(K) defined as the uniform closure of the rational functions with poles outside K. Froelich and Marsalli showed that if the restriction of the measure µ to the boundary of K is discrete then the unit ball of A(K) is strongly precompact, and that if the unit ball of R(K) is strongly precompact then the restriction of the measure µ to the boundary of each component of C\K is discrete. The aim of this paper is to provide three examples that go to clarify the results of Froelich and Marsalli; in particular, it is shown that the converses to both statements are false.

Let (X, d) be a pointed compact metric space, let 0 < α < 1, and let ϕ : X → X be a base point-preserving Lipschitz map. We show that the essential norm of the composition operator Cϕ induced by the symbol ϕ on the Lipschitz spaces lip0 (X, dα) and Lip0(X, dα) is given by the formula kCϕke = lim t→0 sup 0

An algebra of bounded linear operators on a Hilbert space is said to be strongly compact if its unit ball is relatively compact in the strong operator topology. A bounded linear operator on a Hilbert space is said to be strongly compact if the algebra generated by the operator and the identity is strongly compact. This notion was introduced by Lomonosov as an approach to the invariant subspace problem for essentially normal operators. First of all, some basic properties of strongly compact algebras are established. Next, a characterization of strongly compact normal operators is provided in terms of their spectral representation, and some applications are given. Finally, necessary and sufficient conditions for a weighted shift to be strongly compact are obtained in terms of the sliding products of its weights, and further applications are derived., Un álgebra de operadores lineales en un espacio de Hilbert se dice que es fuertemente compacta si su bola unidad es relativamente compacta en la topología fuerte de operadores. Un operador lineal y continuo en un espacio de Hilbert es fuertemente compacto si el algebra generada por el operador y la identidad es fuertemente compacta. Esta noción fue introducida por Lomonosov para estudiar el problema del subespacio invariante para operadores esencialmente normales. En primer lugar, se establecen algunas propiedades básicas de las álgebras fuertemente compactas. Se proporciona después una caracterizacion de los operadores normales fuertemente compactos en términos de su representación espectral y se dan algunas aplicaciones. Finalmente, se obtienen condiciones necesarias y suficientes para que un desplazamiento ponderado sea fuertemente compacto en términos de los productos deslizados de sus pesos. Se proporcionan algunas otras aplicaciones.

An algebra of bounded linear operators on a Hilbert space is said to be strongly compact if its unit ball is relatively compact in the strong operator topology. A bounded linear operator on a Hilbert space is said to be strongly compact if the algebra generated by the operator and the identity is strongly compact. This notion was introduced by Lomonosov as an approach to the invariant subspace problem for essentially normal operators. First of all, some basic properties of strongly compact algebras are established. Next, a characterization of strongly compact normal operators is provided in terms of their spectral representation, and some applications are given. Finally, necessary and sufficient conditions for a weighted shift to be strongly compact are obtained in terms of the sliding products of its weights, and further applications are derived., Un álgebra de operadores lineales en un espacio de Hilbert se dice que es fuertemente compacta si su bola unidad es relativamente compacta en la topología fuerte de operadores. Un operador lineal y continuo en un espacio de Hilbert es fuertemente compacto si el algebra generada por el operador y la identidad es fuertemente compacta. Esta noción fue introducida por Lomonosov para estudiar el problema del subespacio invariante para operadores esencialmente normales. En primer lugar, se establecen algunas propiedades básicas de las álgebras fuertemente compactas. Se proporciona después una caracterizacion de los operadores normales fuertemente compactos en términos de su representación espectral y se dan algunas aplicaciones. Finalmente, se obtienen condiciones necesarias y suficientes para que un desplazamiento ponderado sea fuertemente compacto en términos de los productos deslizados de sus pesos. Se proporcionan algunas otras aplicaciones.

An algebra of bounded linear operators on a Hilbert space is said to be strongly compact if its unit ball is relatively compact in the strong operator topology. A bounded linear operator on a Hilbert space is said to be strongly compact if the algebra generated by the operator and the identity is strongly compact. This notion was introduced by Lomonosov as an approach to the invariant subspace problem for essentially normal operators. First of all, some basic properties of strongly compact algebras are established. Next, a characterization of strongly compact normal operators is provided in terms of their spectral representation, and some applications are given. Finally, necessary and sufficient conditions for a weighted shift to be strongly compact are obtained in terms of the sliding products of its weights, and further applications are derived., Un álgebra de operadores lineales en un espacio de Hilbert se dice que es fuertemente compacta si su bola unidad es relativamente compacta en la topología fuerte de operadores. Un operador lineal y continuo en un espacio de Hilbert es fuertemente compacto si el algebra generada por el operador y la identidad es fuertemente compacta. Esta noción fue introducida por Lomonosov para estudiar el problema del subespacio invariante para operadores esencialmente normales. En primer lugar, se establecen algunas propiedades básicas de las álgebras fuertemente compactas. Se proporciona después una caracterizacion de los operadores normales fuertemente compactos en términos de su representación espectral y se dan algunas aplicaciones. Finalmente, se obtienen condiciones necesarias y suficientes para que un desplazamiento ponderado sea fuertemente compacto en términos de los productos deslizados de sus pesos. Se proporcionan algunas otras aplicaciones.

Sign.: a-h>4

Sign.: [a]-h<4>, Port. con viñeta xil

Respétase a puntuación orixinal, No texto fase referencia a sede vacante do Arcebispado de Santiago de Compostela entre 1820-1822, Tít. tomado do comenzo do texto, Texto asinado por "André Rivera" e "Miguel Benito Cantero y Grande"

Sinaturas: []4, b-h4, Outros