Topics in Complex Analysis and Operator Theory PDF
Preview Topics in Complex Analysis and Operator Theory
Topics in Complex Analysis and Operator Theory DANIEL GJRELA ALVAREZ CRISTOBAL GONZALEZ ENRIQUEZ (eds.) TOPICS IN COMPLEX ANALYSIS AND OPERATOR THEORY DANIEL GIRELA A´LVAREZ CRISTO´BAL GONZA´LEZ ENR´IQUEZ (Eds.) TOPICS IN COMPLEX ANALYSIS AND OPERATOR THEORY UNIVERSIDAD DE MA´LAGA (cid:13)c Los autores (cid:13)c Servicio de Publicaciones e Intercambio Cient´ıfico de la Universidad de M´alaga Imprime: Imagraf Impresoras, S.A. Tel.: 952 32 85 97 ISBN: 978-84-9747-174-9 Dep´osito Legal: MA-172/2007 Disen˜o de cubierta: Ma Luisa Cruz. Servicio de Publicaciones. Universidad de M´alaga v Contents Preface............................................................. vii List of Participants................................................. xi Program............................................................ xv I Articles................................................ 1 A. Aleman A class of integral operators on spaces of analytic functions ...... 3–30 D. Drasin Asymptotic formulas and zero-distribution of certain entire functions ........................................................ 31–40 K.-G. Grosse-Erdmann Dynamics of linear operators ..................................... 41–84 E. Saksman An elementary introduction to Clark measures ................... 85–136 II Research Problems.................................... 137 A. Aleman A class of integral operators on spaces of analytic functions ......139–140 N. Arcozzi, R. Rochberg, and E. Sawyer Some problems on Carleson Measures for Besov-Sobolev Spaces ..141–148 O. Blasco Boundedness of the bilinear Hilbert transform on Bergman spaces 149–152 J. Bonet Interpolating sequences for weighted spaces of entire functions ....153–154 A. Bonilla and K.-G. Grosse-Erdmann A problem concerning the permissible rates of growth of frequently hypercyclic entire functions ......................................155–158 vi S. Ostrovska Eigenvalues and eigenfunctions of the limit q-Bernstein operator ..159–160 D. Vukoti´c An open problem for Toeplitz operators ..........................161–162 vii Preface This volume comprises the conference proceedings of the “Winter School in Complex Analysis and Operator Theory”, held in Antequera, M´alaga, Spain, February 5-9, 2006. This winter school was organized as part of the activities of the Spanish network “Complex Variables, Function Spaces and Operators between them”. This network is currently composed by approxi- mately70researchersof14Spanishuniversitieswhosemainareasofresearch areComplexAnalysisandOperatorTheoryandissupportedbygrantsfrom theSpanish“MinisteriodeEducaci´onyCiencia” (accionescomplementarias MTM2004-21420-E and MTM2006-26627-E). Complex Analysis and Operator Theory are two of the most classical branches of mathematical analysis and they are very closely related. Fur- thermore, they have close links with others such as Harmonic Analysis, Probability, Functional Analysis and Geometry. The interplay between all these branches of mathematics gives rise to very beautiful results and makes our area of research so nice and alive. The winter school was organized with the aim of providing an extensive overview of some of the lines of research followed by the distinct groups which compose the network. It consisted of three four-hours mini-courses, invited lectures of 25 minutes each, and a session on open problems. A number of posters were presented also. Alexandru Aleman delivered a course about integral operators on spaces of analytic functions on the unit disc D, of the form (cid:90) z T f(z) = f(t)g(cid:48)(t)dt g 0 where g is a fixed analytic function on D, which is called the symbol of the operator. These operators appear in a natural way in many problems in complex analysis. In his lectures Aleman discussed various basic properties oftheseoperatorsactingbetweensomecommonspacesofanalyticfunctions on the disc, more precisely between Hardy and weighted Bergman spaces. Most known results about these operators characterize the symbols g such that T acts as a bounded, compact or Schatten-class operator between g given spaces as above. After presenting some of these, Aleman turned to more specific questions like the spectrum and invariant subspaces for such operators. These problems turn out to be quite difficult in the general case viii and for that reason certain special cases which are of interest in their own right were considered. Karl Grosse-Erdmann delivered a course on dynamics of linear operators. In recent years a theory of linear dynamical systems has started taking shape. Building upon the simple notion of a hypercyclic operator, that is, an operator with a dense orbit, further concepts have been introduced and studied: chaotic operators, weakly mixing and mixing operators, and, most recently, frequently hypercyclic operators. In his course, Grosse-Erdman gave an introduction to this emerging theory, stating its main definitions and presenting its main results. He, in particular, applied the theory to operators on spaces of analytic functions. Next, he concentrated on recent investigations of F. Bayart and S. Grivaux who have applied methods from ergodic theory to the study of linear dynamical systems. The link between the two areas is provided by the notion of a Gaussian measure that is in- variant for a given operator; properties of the invariant measure determine the dynamic behaviour of the operator. Eero Saksman delivered the third mini-course which was entitled “Clark measures, composition operators, and related questions”. Associated with an analytic self-map ϕ of the unit disc there is a family {µ : α ∈ ∂D} of α Borel measures on the boundary of the unit circle. The measures µ are α called the spectral measures, or Clark measures (or Alexandrov-Clark mea- sures), associated to the function ϕ. These measures have turned out to be extremely useful in many questions of complex analysis or operator theory. Saksman’s purpose in these lectures was to investigate Clark measures and some of their applications in operator related function theory, starting from basic notions. Especially, he explored their central role in the theory of composition operators. He also investigated in more detail the connection between the Nevanlinna counting function of ϕ and the Clark measures, starting from Alexandrov’s theory of values distribution of inner functions on the boundary ∂D. The invited talks dealt with recent results in a variety of lines of research within our area including: Carleson measures on functions spaces, problems of interpolation by analytic functions, inner functions, complex dynamics, ix aspects of the theory of distinct type of operators acting on abstract Ba- nach spaces and of some concrete operators acting on spaces of analytic functions, etc. This volume contains papers by Alexandru Aleman, Karl Grosse-Erdman and Eero Saksman which are very closely related to the courses they deliv- ered in the winter school. We include also a paper by David Drasin on the relation between the zero-distribution of a certain entire function f and the asymptoticformulaforlog|f|fromanewperspective: Drasinviewsthefinal asymptoticformoflogf asthefundamentalobject, andrelayonelementary potential theory to use this form to determine the asymptotic distribution of the zeros. In addition to this, we include in this volume a number of research prob- lems in distinct topics of our area which were presented in the session on open problems of the winter school. Enough references are included so that thereadercanaccesstherecentliteraturerelatedtoeachparticularproblem. These problems will be published also in the section “Research Problems” of the web page of our network http://www.uma.es/investigadores/grupos/cfunspot/ In fact, we shall inaugurate this section of the web page with these prob- lems. Subsequently,otherproblemswillbeaddedandwedoexpectthatthis will help us to be able to turn our web page into a useful tool for research in Complex Analysis and Operator Theory, as well as a meeting point for researchers working on this field or on any other related topic. WewishtothankAlexandru, KarlandEerofortheexcellentcoursesthey delivered. Weareindebtedalsotoallthespeakerswhopresentedreallyvery good talks and, in general, to all the participants. The winter school was a success because of them. We have to express also our gratitude to: • distinct institutions such as: El Departamento de An´alisis Matem´ati- co y el Vicerrectorado de Investigaci´on de la Universidad de M´alaga, La Junta de Andaluc´ıa y El Ministerio de Educaci´on y Ciencia which sponsored partially the event, • our colleagues Nancho A´lvarez, Auxi M´arquez and Antonio Jim´enez who helped us in the organization of the meeting,
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