Dilation theory has been particularly successful in the study of contractions acting on a Hilbert space H. The aim of this lecture is to associate a dilation curve with any operator T in B(H), to give properties of this curve and its applications in several areas of operator theory. We show how it allows us to make a Julia analysis and a Harnack ordering for operators. This dilation curve also intervenes into estimates for hyperbolic metric and could be used more generally to obtain sharpened inequalities involving operators. In particular, we give optimal estimates for the norm ||f(T₁)X − Xf(T₂)|| where T₁, T₂ and X are three operators in B(H) and f belongs to special function classes.

For a bounded normal operator N and a bounded Borel function f:σ(N) → C one defines f(N) as the integral ∫_σ(N) f(t) dEN(t) where EN is the spectral measure of N. Using the functional calculus defined above, many classical properties of scalar functions (such as being a monotone, convex, Lipschitz or Hölder function) are transferred into the operator context. For example, a (Borel) function u:∆ → R (where ∆ ⊂ R is an interval) is operator monotone iff for any two bounded selfadjoint operators A and B acting on a common Hilbert space, whose spectra are contained in ∆ one has u(A) ⩽ u(B). There is a fascinating theorem due to Loewner which fully characterizes all operator monotone functions. The talk will be devoted to the proof of Loewner’s theorem based on Choquet’s simplices. If time permits, some information on operator Hölder and operator Lipschitz functions will be given. Surprisingly, these two similar classes totally differ in their operator versions.

Przedstawię podstawy tak zwanej nieprzemiennej topologii, czyli dziedziny zajmującej się badaniem "przestrzeni kwantowych", tj. obiektów kategorii dwoistej do kategorii (być może nieprzemiennych) C*-algebr. Podkategoria obiektów dwoistych do przemiennych C*-algebr odpowiada dokładnie lokalnie zwartym przestrzeniom z odwzorowaniami ciągłymi jako morfizmami. W tym kontekście wprowadzę pojęcie ciągłej kwantowej rodziny odwzorowań. Omówię problem istnienia kwantowej przestrzeni wszystkich odwzorowań z jednej przestrzeni kwantowej do drugiej i pokażę, że jeśli taka przestrzeń istnieje, to posiada bogatą strukturę "półgrupy kwantowej". Przedstawię kilka przykładów takich obiektów.

Orthogonal polynomials show up in many areas of modern mathematics. Some of them are eigenfunctions of differential operators. Some other play a role of matrix coefficients of representations of matrix groups. General orthogonal polynomials are always associated with special symmetric linear operators called Jacobi matrices. These matrices can be either bounded or unbounded. The aim of the lecture is to give introduction to orthogonal polynomials and show connections to Jacobi matrices. I expect the audience is familiar with basic theory of bounded linear operators. Some elements of the theory of unbounded operators will be used as well, and explained on-line if necessary.

Endre Szemerédi has proved in 1974 that any set of natural numbers having strictly positive upper Banach density contains arithmetic progressions of arbitrary length (the Erdös-Turán conjecture). His proof is combinatorial and works via a graph-theoretic problem. In 1977 Hillel Furstenberg proved a multiple recurrence theorem for transformations preserving a probability measure, which implies Szemerédi’s theorem. Part of Furstenberg’s results holds also in the framework of dynamical systems on general operator algebras. I intend to discuss the operator theoretical methods involved in the investigation of multiple recurrence properties of dynamical systems on operator algebras with final land on Szemerédi’s theorem.

A simple proof of the Banach fixed-point theorem is presented. Then it is presented how the theorem can be applied to generating pictures (i.e. in fractal compression). Moreover a bit of information about fractals is given. Finally a little (non-obvious) digression and considerations are presented how fractals describe Nature.

In Topology and Functional Analysis, there are many fixed point theorems (the most famous among them are the fixed point theorems of Banach, Brouwer, and Schauder). They are often very nontrivial and useful. In the talk I will present some versions, proofs and applications of the Markov - Kakutani (simultaneous) fixed point theorem, whose standard statement reads as follows. Let K be a nonempty compact convex subset of a locally convex space. Then every commuting family of continuous affine maps of K into itself has a common fixed point.

We will pressent few facts concerning structure of point spectra of bounded linear operators on a Banach space. Among other things, the result that in reflexive spaces they always are F_σ, and in Hilbert spaces these are exactly F_σ sets.

Gurarii space is a separable Banach space G satisfying the following condition: (*) Given finite-dimensional Banach spaces X ⊆ Y, given ε > 0, given an isometric embedding f: X → G there exists an injective linear operator g: Y → G extending f and satisfying |g|·|g^-1| ≤ 1 + ε. This space was found by Gurarii in 1966. Its uniqueness up to isometry was proved by Lusky in 1976. We shall present a category-theoretic description of the Gurarii space, obtaining simpler proofs of some known properties of this space.

We will present a variational rinciple due to A. B. Antonevich and A. V. Lebedev which expresses the spectral radius of weighted composition operators (aTf)(x)=a(x)f((x)), f ∈ F(X), acting in various function spaces F(X). Moreover we will explain equivalence of this result with a similar formula obtained independently by A. Kitover for spectral radius of weighted automorphism of the function algebra C(X). This motivates an abstract C*-algebraic approach to the theory of the operators under consideration.

I'll present some basic theorems of Banach Space Geometry escpecially Rodé's theorem concerning weak* closure of convex compact set. Main tool used in proof will be the Bischop-Phelps theorem which we will also proof during talk.

Consider the following famous conjectures about derivations on Banach algebras:(C1) every derivation on a Banach algebra has a nilpotent separating ideal, (C2) every derivation on a semiprime Banach algebra is continuous, (C3) every derivation on a prime Banach algebra is continuous, (C4) every derivation on a Banach algebra leaves each primitive ideal invariant. Conjectures (C1), (C2) and (C3) are still open, even in the case of commutative Banach algebras. Assertion (C4) is usually referred to as the noncommutative Singer-Wermer conjecture. It is true in the commutative case. We will present some remarks about relations between these conjectures.

The standard way to analyze a broad class of complex 1D quantum system in physics is so called Density Matrix Renormalization Group (DMRG). This is a family of effective numerical techniques. A natural representation of such a system arises within DMRG: it is so called Matrix Product States (MPS) representation. It has a large potential for mathematical research. However, not much of investigation by mathematicians was performed neither for DMRG, nor for MPS. I my talk I will give a brief review of physical background (what are quantum complex systems), mathematics of MPS (I will try to summarize all of the research done for MPS so far, including some statements about when can we describe quantum complex systems by DMRG/MPS efficiently). In the last part I will present my results on application of MPS to perform a spectral analysis of a complex quantum system. I would like to include detailed results in a poster (supplement to a talk).

The fifth problem on the famous list of 23 mathematical questions published in 1900 by David Hilbert was the following (roughly speaking): if a topological group is locally homeomorphic to a Euclidean space, is it necessarily diffeomorphic to this space? The case of compact groups was solved by John von Neumann in 1933 using Peter-Weyl theorem. During the talk we will see the proof of the result of Peter and Weyl by studying the structure of compact self-adjoint operators acting on a Hilbert space. We will also sketch the rest of the proof given by von Neumann and mention how this result becomes a foundation of Fourier analysis in more general setting.

The lecture will deal with applications of the Hahn-Banach theorem in welfare economics, a branch of mathematical economics that focuses on how the optimal allocation of resources affects social welfare. We will present the second fundamental theorem of welfare economics and its proof.

Let us denote by H a complex Hilbert space. We will consider the following operator spaces: - B(H) is the complex linear space of all bounded operators on H, - H(H) is the real linear space of all self-adjoint bounded operators on H, - B_K(H) is the complex linear space of all operators on H which have the form α id_H + K, where α is a scalar coefficient and K is a compact operator, - H_K(H) = B_K(H) cap H(H) (it is a real linear space). We denote by σ(X) the spectrum of an operator X in B(H), and define the numerical range of X by W(X) = {<Xv,v> : v ∈ H, <v,v> = 1}, where <⋅,-> is the inner product on H. The main goal of the talk is to present a proof (based on the original paper) of the following theorem [1]. Theorem 1. [Chi-Kwong Li, Rodman, v Semrl] Let V be one of the above operator spaces and let ε: V→V be a surjective linear map. Then the following conditions are equivalent (1) ε maps the set of self-adjoint elements into itself, and σ(ε(X)) = σ(X) for all self-adjoint X in V, (2) ε is continuous (in the norm topology), maps the set of self-adjoint elements into itself, and is such that σ(ε(X)) = σ(X) for all orthogonal projections X in V, (3) W(ε(X)) = W(X) for all X in V, (4) W(ε(X)) = W(X) for all X in V, (5) there is a unitary operator U in B(H) such that varphi is of the form X ↦ UXU* or X ↦ UX^tU*, where X^t is the transpose of X with respect to some fixed orthonormal basis in H. At the end of the talk we will present a generalization of the theorem [2]. The bibliography: [1] Chi-Kwong Li, L. Rodman, P. v Semrl, Linear Maps on Selfadjoint Operators. Preserving Invertibility, Positive Definiteness, Numerical Range, Canad. Math. Bull. 46 (2): 216-228 (2003). [2] S. Clark, Chi-Kwong Li, J. Mahle, L. Rodman, Linear Preservers of Higher Rank Numerical Ranges and Radii, Linear Multilinear Algebra 57 (5): 503-521 (2009).

The operator A is called a Riesz-spectral operator if it possesses a Riesz basis consisting of eigenvectors (a Riesz basis is some generalization of the orthonormal basis), the corresponding eigenvalues are all simple, and the spectrum is totally disconnected. Riesz-spectral operators are connected with many problems in partial differential equations. They also have quite simple representation and allow nice characterizations of many properties such as generation of C₀ semigroup, invariant subspaces, exponential stability. In this talk basic facts concerning Riesz basis and properties of Riesz-spectral operators with some examples will be presented. References [1] R. F. Curtain, H. J. Zwart. An Introduction to Infinite Dimensional Linear Systems Theory. Springer, New York, 1995. [2] I. C. Gohberg, M. G. Krein. Introduction to the Theory of Linear Non- selfadjoint Operators in Hilbert Space. American Mathematical Society, Providence, 1969. [3] B.Z. Guo, H. J. Zwart. Riesz Spectral Systems. Memorandum No. 1594 , Faculty of Mathematical Science,University of Twente, November 2001.

The following question was asked in Banach's book Théorie des opérations linéaires: is l₂ the only infinite-dimensional Banach space (up to isomorphic) which is isomorphic to all its infinite-dimensional subspaces? During the lecture I will present an important special case of Gowers' dichotomy theorem, which together with the Komorowski and Tomczak-Jaegermann's theorem, gives a positive answer to above question and thus a new isomorphic characterization of separable Hilbert spaces. Dichotomy theorem, whose proof is based on application of Ramsey theory to Banach spaces, is wonderful illustration of permeation of mathematical branches, which are not as remote from each other as they seem to be and thus confirm us in its inner coherence.

This is joint results with K. Zajkowski. We derive a relationship between the Legendre-Fenchel transform of the spectral exponent of weighted composition operators acting in L^p-spaces and the Legendre-Fenchel transform obtained for their analytic functions. We establish the variational principle for the spectral exponent of analytic functions of weighted composition operators.