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Minicourses on operator theory and complex analysis

In the week of 6-10 May 2013, the Doctoral Programme organizes two minicourses (8 hours each) on operator theory and complex analysis aimed at doctoral students and advanced undergraduates in mathematics as well as postdocs and faculty members.

The minicourses will take place at the Department of Mathematics and Statistics of the University of Helsinki. It is located in the Exactum building (Gustaf Hällströmin katu 2b) on the Kumpula Campus.

1. Operators on Model Spaces

Lecturer: William Ross (University of Richmond, USA)

Lecture notes (exercises added on 10 May)

Contents: Inner functions - Hardy spaces - Model spaces - Clark theory - Aleksandrov's disintegration theorem - Truncated Toeplitz operators - Complex symmetric operators

2. Spaces of Analytic Functions, Geometry of Domains, and Superposition Operators

Lecturer: Dragan Vukotić (Universidad Autónoma de Madrid, Spain)

Lecture notes (corrected and final version added on 20 June)


  1. A quick review of univalent functions: growth theorem, Koebe 1/4 theorem. Distance to the boundary of the image domain.

  2. A quick review of the hyperbolic metric in the disk.

  3. Hardy spaces of the disk; Blaschke products and Riesz factorization. Bergman spaces. Domains with rectifiable boundary and the isoperimetric inequality.

  4. Whitney decomposition of planar domains. A topological theorem on simply connected subdomains of arbitrary domains of finite area.

  5. Growth estimates and the boundary distribution function for the Dirichlet space. Theorems of Beurling and Chang-Marshall. An approach via Green's formula.

  6. Dirichlet type spaces. Conformally invariant spaces: Bloch and Besov. Membership of conformal maps in conformally invariant spaces in terms of geometric properties of domains.

  7. A quick review of entire functions of exponential growth: order and type.

  8. (Non-linear) superposition operators between different pairs of spaces of analytic functions: Bergman to Bergman, Bloch to Bergman, Dirichlet type to Dirichlet type, etc.



Mon 6

Tue 7

Wed 8

Thu 9

Fri 10





Ascension Day

no lectures



lunch break

lunch break

lunch break

lunch break






All lectures are held in the lecture room D123 (first floor).

Conference dinner on Wednesday at 19 o'clock at Restaurant Kaarna (map, menu in Finnish, menu in English). Participants are expected to pay for themselves.


Although there is no registration fee for attending the event, participants coming from outside the University of Helsinki are asked to notify either of the organizers by email. A limited number of short talks by the participants can be included; please contact the organizers if you wish to give such a talk.

The Doctoral Programme may provide financial support for travel and accommodation costs to doctoral students who work at one of the Finnish departments participating in the programme. Please contact Hans-Olav Tylli for details.

The participants are requested to arrange their accommodation themselves. Here are links to just a few hotels that might be considered:


Visit Helsinki: City of Helsinki official website

Suomenlinna Fortress


Hans-Olav Tylli,
Pekka Nieminen,

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