Resumen de Sobre espacios y álgebras de funciones holomorfas

Pablo Sevilla-Peris

• español

  La Tesis Sobre espacios y ¿algebras de funciones holomorfas se estructura en tres cap¿ýtulos diferentes.

  En cada uno de ellos se aborda un problema diferente.

  El primer cap¿ýtulo se dedica al estudio de los operadores de composici¿on. La idea original es bastante sencilla y natural. Tomamos el disco unidad complejo, que denotamos D, y una funci¿on holomorfa : D -! D. Con esto se define un operador f 7! f donde f : D -! C es una funci¿on holomorfa.

  Consideramos el caso en que en lugar de D tenemos B, la bola unidad de un espacio de Banach y el operador est¿a definido entre espacios ponderados de funciones holomorfas. Generalizamos algunos resultados relativos a la continuidad y la compacidad del operador dados por Bonet, Doma¿nski, Lindstr ¿om and Taskinen para el caso unidimensional.

  En en el segundo cap¿ýtulo se estudia la teor¿ýa espectral en ¿algebras lmc. La teor¿ýa espectral cl¿asica para elementos de un ¿algebra unitaria cualquiera ha sido ampliamente estudiada y desarrollada. Durante la d¿ecada de los 1930 Gelfand desarroll¿o un trabajo en el que relacionaba la teor¿ýa espectral en ¿algebras de Banach conmutativas con los homomorfismos de ¿algebras continuos.

  Haciendo uso del espectro definido por Harte en los anos 1970 definimos un espectro vectorial para elementos de un producto tensorial. Tomamos A un ¿algebra con unas ciertas propiedades y E un espacio localmente convexo, para cada T 2 A T E definimos y estudiamos un espectro (T) E. Los primeros pasos en esta direcci¿on fueron dados por Dineen, Harte y Taylor para espacios y ¿algebras de Banach.

  En el ¿ultimo cap¿ýtulo se estudia el cotipo 2 de ciertos espacios de polinomios. Los or¿ýgenes del estudio del tipo y el cotipo se remontan la la d¿ecada de 1930, en estudios de Orlicz. En la d¿ecada de 1970 se formalizaron los dos conceptos.

  Un resultado probado por Dineen en 1995 tiene como consecuencia inmediata que si E es un espacio de Banach de dimensi¿on infinita entonces P(mE) no tiene cotipo 2. Si X es un espacio de Banach de sucesiones y Xn son determinados subespacios finitodimensionales, la sucesi¿on (C2(P(mXn)))n debe tender a 1. En este cap¿ýtulo damos una descripci¿on asint¿otica de esta divergencia.

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• English

  The Thesis with title Sobre espacios y ¿algebras de funciones holomorfas has three dierent chapters.

  Each one of them is devoted to a dierent subject. The first chapter studies composition operators, the second one studies spectral theory on lmc-algebras and the third and last chapter is devoted to the study of cotype 2 of spaces of polynomials on Banach sequence spaces.

  The starting idea of composition operators is simple and a very natural question. Consider D the open unit disc of C and a holomorphic map : D -! D. If f : D -! C is a holomorphic function, we can compose f and try to analyze what happens when we let the f vary; in other words we define an operator between spaces of holomorphic functions and we want to study what properties does this operator have (continuity, compactness, . . . ). This obviously depends on which are the spaces considered.

  Bonet, Doma¿nski, Lindstr¿om and Taskinen studied the situation for weighted spaces.

  Our aim is to generalize some of the results in that paper when we consider B instead of D and de- fine the composition operator between two weighted spaces of holomorphic functions between Banach spaces. We obtain characterizations for such a composition operator to be continuous and compact.

  We end the chapter by considering the situation when we have a countable family of weights instead of one single weight.

  Spectral theory was developed at the beginning of the 20th century and is now a classical subject.

  During the 1930s Gelfand developed a theory, in which the relationship between spectra and the multiplicative linear functionals or the maximal closed ideals of the algebra is outlined.

  The spectra of a family of elements was developed during the 1950s by Arens, Calder¿on, Waelbroeck and Zygmund in the commutative case. Left, right and joint spectra for families of elements in a noncommutative algebra were defined by R.E. Harte during the 1970s. L. Waelbroeck defined a vector spectrum in the early 1970s and the spectrum of elements of A X, A a commutative unital Banach algebra and X a Banach space.

  S. Dineen, R.E. Harte and C. Taylor, developed a vector Gelfand theory for elements in A X, where A is a Banach algebra, X a Banach space and a uniform tensor norm and generalized the Waelbroeck spectrum. Using the spectrum defined by Harte, they defined a left spectrum when A is not commutative. In a series of three papers they presented their results.

  We generalize some of their results to lmc algebras, Q-algebras and locally convex spaces. We obtain results concerning left invertibility of continuous mappings and of holomorphic germs with values in a non-commutative algebra.

  The study of type and cotype of Banach spaces started in the early 1970s, but its origins go back to the 1930s. Orlicz, while studying the unconditional convergence of a series of functions using Khinchins inequality established the first type-cotype style inequality. In 1968 Kahane proved his generalization of Khinchins inequality and these ideas were revisited and used for the study of the relations of strong p-summability and unconditional summability. But in 1972 Kwapie¿n proved that Hilbert spaces are the only Banach spaces that simultaneously have type 2 and cotype 2, although he did not explicitly use those names. Shortly after that achievement the concepts of type and cotype were formulated and widely used.

  In 1995 Dineen showed that if E is an infinite-dimensional Banach space, then `1 is finitely representable in P(mE) for all m 2. This in particular means that P(mE) does not have cotype 2. If X is a Banach sequence space (for example `p with 1 p 1) and we denote by Xn the subspace spanned by the first ek vectors, k = 1, . . . , n, this implies that the sequence (C2(P(mXn)))n must tend to 1.

  We give asymptotical descriptions of this divergence.