Advanced Course in Operator Theory and Complex Analysis (1st. 2004. Sevilla)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/61375

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How large is a Riemann surface: the type problem
(Editorial Universidad de Sevilla, 2005) Drasin, David; Montes Rodríguez, Alfonso
The uniformization theorem asserts that a simply-connected non-compact Riemann surface S is conformally equivalent to precisely one of the unit disk D or the finite complex plane C. While this result (nearly a century old) closes one chapter in the theory of analytic functions of one complex variable, it opens another: given a surface S described in some explicit manner, determine from intrinsic considerations which of the conformal types S is. While this subject reached a zenith of activity in the 1930s, recent developments and the availability of new tools suggest a resurgence of interest.
Volterra operators on spaces of analytic functions - a survey
(Editorial Universidad de Sevilla, 2005) Siskakis, Aristomenis; Montes Rodríguez, Alfonso
We give a short and selective account of results known about operators of the form Vg(f)(z) = 1/z Z z 0 f(ζ)g0 (ζ) dζ, where g is analytic on the disc and the operator Tg = zVg acts on spaces of analytic functions.
Analytic contractions and boundary behaviour - an overview
(Editorial Universidad de Sevilla, 2005) Aleman, Alexandru; Richter, Stefan; Sundberg, Carl; Montes Rodríguez, Alfonso
Composition operators on the weighed Bergman-Nevanlinna classes
(Editorial Universidad de Sevilla, 2005) Sharma, Somdatt D.; Sharma, Ajay K.; Montes Rodríguez, Alfonso
In this paper we use an α-Carleson measure and a vanishing Carleson measure to characterize bounded and compact composition operators on weighted Bergman-Nevanlinna spaces.
Invariant subspaces of translation semigroups
(Editorial Universidad de Sevilla, 2005) Power, Stephen C.; Montes Rodríguez, Alfonso
In these lectures I shall give an account of some recent results and open problems relating to subspaces of square integrable functions on the real line which are jointly invariant for a pair of semigroups of unitary operators. These semigroups are quite fundamental, namely, translation semigroups, Fourier translation semigroups, and dilation semigroups. A celebrated theorem of Beurling gives a description of the closed subspaces that are simply invariant for translations (or Fourier translations) and as a result these subspaces are in bijective correspondence with the set of all unimodular functions. In contrast, subspaces that are invariant for two of these semigroups turn out to be finitely parametrised by a family of specific unimodular functions. I shall indicate how one goes about the identification of these sets of invariant subspaces and how, with the natural topology, they are identifiable as Euclidean manifolds. Also I shall discuss aspects of the relatively novel nonselfadjoint operator algebras that are associated with them. These algebras are generated by two non-commuting copies of H∞(R). To identify the topology on the set of invariant subspaces it turns out that one needs to establish some essentially function theoretic assertions, in which a limit of a sequence of (projections onto) purely invariant subspaces is a particular reducing subspace (projection). We sketch below how one can obtain such “strange limits”. I would like to thank Alfonso Montes-Rodríguez for the opportunity to present these lectures at the University of Seville to an ideal audience composed of a good mix of old hands and young minds.
On spectral structure of bounded linear operators on reflexive Banach spaces
(Editorial Universidad de Sevilla, 2005) Shkarin, Stanislav A.; Smolyanov, Oleg G.; Montes Rodríguez, Alfonso
A descriptive characterization of point, continuous, and residual spectra of operators acting on a separable Hilbert space is obtained. The possible point spectra of bounded linear operators acting on lp, 1 < p < ∞ are characterized.
A parametrization for the symbols of a Hankel type operator
(Editorial Universidad de Sevilla, 2005) Bermudo Navarrete, Sergio; Marcantognini, Stefania A. M.; Morán, María Dolores; Montes Rodríguez, Alfonso
Hankel operators and their symbols, as generalized by V. Pták and P. Vrbová, are considered. In this more general framework, a linear operator X from a Hilbert space H1 to a Hilbert space H2 is said to be a Hankel operator for given contractions T1 on H1 and T2 on H2 if, and only if, XT ∗ 1 = T2X and X satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations V1 of T1 and V2 of T2. A Hankel symbol of X is a dilation Z of X, with a certain norm constraint, such that ZV ∗ 1 = V2Z. The boundedness condition imposed to X has revealed to be essential, indeed necessary and sufficient, for X to admit Hankel symbols. As for a description of the symbols of X, this work provides a parametric labeling of all of them by means of Schur like formula. As a by-product, a new proof of the existence of Hankel symbols is obtained. The proof is established by associating to X, T1 and T2 a suitable isometry V so that there is a bijective correspondence between the symbols of X and the family of all minimal unitary extensions of V.
On the growth of Hardy and Bergman norms of functions in the Dirichlet space
(Editorial Universidad de Sevilla, 2005) Vukotic, Dragan; Montes Rodríguez, Alfonso
We review the Chang-Marshall inequality of Moser-Trudinger type for the Dirichlet space. We then use a weaker version of this result to derive a sharp asymptotic estimate for Hardy and Bergman norms of a Dirichlet function for large exponents.
Banach algebras with trivial cohomology
(Editorial Universidad de Sevilla, 2005) El Harti, Rachid; Montes Rodríguez, Alfonso
A plausible conjecture is that Banach algebras with trivial cohomology have to be semisimple and finite dimensional. In this paper, we show that a Hermitian Banach ∗-algebra has trivial cohomology if and only if it is ∗-isomorphic to a finite direct sum of full matrix algebras.
Composition operators with linear fractional smbols and their adjoints
(Editorial Universidad de Sevilla, 2005) Martín Gómez, María José; Montes Rodríguez, Alfonso
We characterize all linear fractional maps of the disk into itself in terms of their coefficients. We also prove the formula for the adjoint of a composition operator with a linear fractional symbol acting on the quotient Dirichlet space due to Gallardo and Montes by a method different from theirs.
Toeplitz operators and division by inner functions
(Editorial Universidad de Sevilla, 2005) Girela Álvarez, Daniel; González Enríquez, Cristóbal Miguel; Peláez Márquez, José Ángel; Montes Rodríguez, Alfonso
A subspace X of the Hardy space H1 is said to have the K-property if for any ψ ∈ H∞, the Toeplitz operator Tψ maps X into itself. This in turn implies that X also has the f-property. This means that h/I ∈ X whenever h ∈ X and I is an inner function with h/I ∈ H1. In this survey paper we present a list of subspaces of H1 that have or have not the f- or K-property, showing some of the different techniques and methods used in the subject.
Operators on Lp and the role of the image of the unit ball
(Editorial Universidad de Sevilla, 2005) Romero Moreno, María del Carmen; Universidad de Sevilla. Departamento de Análisis Matemático; Montes Rodríguez, Alfonso; Universidad de Sevilla. FQM260: Variable Compleja y Teoría de Operadores
Let p ∈ [1, +∞] such that its conjugate exponent q is not an even integer and let T be an operator defined on Lp(λ) with values in a Banach space. In this note we discuss how the image of the unit ball determines whether T belongs to some classes of operators such as operator ideals or the class of representable operators. We also study the monotonicity of these properties, proving that a Banach space is Cisomorphic to a subspace of an Lq space if and only if the representability of every operator on Lp is monotone with respect to the image of the unit ball.

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