
Discrete harmonic analysis seminar  past schedule for the year 2008/2009:Current page of the seminar is available here. 
Kwadratowe harnessy to procesy stochastyczne mające liniową warunkową wartość oczekiwaną i kwadratową warunkową wariancję przy warunkowaniu przez przeszłość i przyszłość łącznie. Przykłady kwadratowych harnessów to: proces Wienera, proces Poissona, proces Dirichleta, wolny proces Wienera, procesy qgaussowskie, proces bipoissonowski. Problem, którym się zajmujemy, to próba kompletnego opisu tej klasy procesów stochastycznych. Okazuje się, że są to zawsze procesy markowskie, co więcej są one wyznaczone jednoznacznie przez 5 stałych rzeczywistych (jedną z nich jest q). Stałe te są parametrami równania qkomutacyjnego wiążącego dwie części macierzy Jacobiego ortogonalnych wielomianów martyngałowych rozważanego procesu. Macierz Jacobiego jest funkcją afiniczną czasu i wspomniane dwie części to odpowiednie współczynniki macierzowe tej właśnie funkcji. W taki schemat po odpowiednim przeparametryzowaniu zanurzyć można szeroką klasę wielomianów AskeyWilsona. Miara ortogonalizująca te wielomiany jest rozkładem jednowymiarowym odpowiedniego kwadratowego harnessa. Wyniki, o których będę mówił, zostały otrzymane wspólnie z W. Brycem i częściowo z W. Matysiakiem.
I will discuss asymptotic properties of sums of matricially free random variables. The concept of matricial freeness can be viewed as a generalization of both freeness of Voiculescu and monotone independence of Muraki. The main idea is that we consider arrays of random variables. In particular, the `rows' of square arrays lead to free random variables whereas the `rows' of lower (upper) triangular arrays lead to monotone (antimonotone) independent random variables. At the same time, the sums of all random variables in the given array, called a random pseudomatrix, have the same asymptotics as a Gaussian random matrix if we assume (in both cases) that the variance matrix is symmetric and has blockidentical variances. The same feature is exhibited by the blocks of both objects.
We will focus this talk on the study of a random matrix model describing the output of a specific input state (the Bell state) under well chosen random quantum channels. This matrix model is quite new from the point of view of random matrix theory and has many interesting properties. In particular, under appropriate scalings we can compute its asymptotic eigenvalue distribution and provide new examples and counterexamples for various additivity conjectures in quantum information theory.
A link between properties of quantized chaotic systems and random matrices will be reviewed. Statistical properties of periodically driven quantum chaotic systems described in a finite dimensional Hilbert space H_N can be described by circular ensembles of random unitary matrices. To describe the effect of a possible interaction of the system in question with an environment one needs to work with density operators, which are Hermitian, positive and normalized. Discrete time evolution of a density matrix can be represented by socalled quantum operation (completely positive, trace preserving map). We introduces several ensembles of random operations, discuss algorithms to generate them and investigate spectral properties of the corresponding superoperators with spectrum consisting of N^2 eigenvalues inside the unit disk. A quantum analogue of the FrobeniusPerron theorem concerning the spectrum of stochastic matrices is formulated. Obtained predictions for random operations are compared with spectral properties of quantized chaotic systems, interacting with an environment.
I will present a new proof of the hypercontractivity inequalities for holomorphic algebras generated by elements fulfilling Canonial Anicommutation Relations (CAR).
Let \Gamma denote a noncommutative free group, and let \Omega stand for its boundary. We construct a large class of unitary representations of \Gamma. This class contains many previously studied representations, and is closed under several natural operations. Each of the constructed representations is in fact a representation of \Gamma\ltimes_\lambda C(\Omega). As representations of \Gamma\ltimes_\lambda C(\Omega), they are all irreducible. As representations of \Gamma, each of them is either irreducible, or the direct sum of exactly two irreducible, inequivalent \Gammarepresentations.
There will be a short introduction to the basic concepts of the talk, namely spherical functions and HerzSchur multipliers. The main focus of the talk is a criterion for determining when a radial function on a nonabelian free group is a HerzSchur multiplier, together with an explicit formula for the HerzSchur norm. This result, obtained by HaagerupSzwarc, has remained unpublished but is to appear as part of a joint paper. As an application of this result one can find closed expressions for the HerzSchur norm of the spherical functions on the nonabelian free groups, which extends previous work by PytlikSzwarc. Haagerup used these formulas to show that there are HerzSchur multipliers on the nonabelian free groups which are not coefficients of uniformly bounded representations. This has remained unpublished, but an alternate proof has since been published by Pisier. Actually, a generalization can be found for countable discrete groups, using that such groups are amenable if and only if the HerzSchur multipliers coincide with the FourierStieltjes algebra (due to Bozejko) and that each HerzSchur multiplier can be realized as the coefficient of a (not necessarily uniformly bounded) representation (due to BozejkoFendler). Finally, we will discuss various extensions to (some) Lie groups and general locally compact groups.
Rozważamy zanurzenie grupy transformacji odwracalnych odcinka [0,1] w algebrę operatorów ograniczonych na przestrzeni Orlicza. Dowodzimy, że włożenie to zachowuje działanie grupowe wtedy i tylko wtedy, gdy przestrzeń Orlicza jest przestrzenią L^p dla pewnego 1\le p<\infty.
We compute moments and free cumulants of the measure \rho_t:=\pi_t\boxtimes\pi_t, where \pi_t denotes the free Poisson law with parameter t>0. We also compute free cumulants of the symmetrization of \rho_t. Finally we introduce free symmetrization of a probability measure on \mathbb{R} and provide some examples.
We expand Chebishev polynomials and some of its linear combination in linear combinations of qHermite and Al SalamChihara polynomials. We use these expansions to expand qNormal and related densities into infinite series of Chebishev polynomials and thus study probabilistic properties of these distributions including efficient simulation. The analysis includs also case q>1. Then distributions are discrete. In the case of conditional normal distribution the support consists of zeros of some AlSalamChihara polynomilas. We find those zeros. We use this result to prove the existence of stationary random fields with linear regressions and thus close an open question posed by W. Bryc et al.. We prove this result by describing a discrete 1 dimensional conditional distribution. As a by product we generalize to qseries case, a well known formula (x+y)^{n}=\sum_{i=0}^{n} \binom{n}{k}i^{k}H_{nk}\left( x\right) H_{k}\left( iy\right) , where H_{k}\left( x\right) denotes kth Hermite polynomial.
In this talk, we present a way to compute the completely bounded multiplier norm of the characteristic function of the set of generators in the free group
I give the classification of families of orthogonal polynomials with ultraspherical type generating functions. This gives identities where a generalized Stieltjes transform of some probability distribution is expressed as the geometric mean of two Stieltjes transforms on the one side and as a powered Stieltjes transform of a probability distribution on the other side. Using Markov transforms, we get a family of symmetric probability distributions interpolating between the arcsine and the Wigner distributions. As a matter of fact, interesting, not yet solved, problems emerge: finding a convolution operation that interpolates between the rconvolution defined by M. Bozejko for r=1/2 and r=1, find a parallel to Young diagrams for which this probability distribution is the rescaled limiting shape.
We will consider extremal elements of the cone of all positive maps acting between B(K) and B(H) where K and H are Hilbert spaces. We will show that every positive map with the property that rank \phi(P)\leq 1 for any onedimensional projection P is a rank 1 preserver. It allows to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2positive turns out to be automatically completely positive. Finally we get the same conclusion for such extremal positive maps that rank \phi(P)\leq 1 for some onedimensional projection P and satisfy the condition of local complete positivity. It allows us to give a negative answer for Robertson's problem in some special cases.
We will review wellknown techniques and a new theorem that permit to associate a Toeplitz Schur multiplier to a Fourier multiplier, and vice versa, and provide some applications.
Gcompact abelian group, \Gammaits dual, S\subset\Gamma an infinite Sidon set; \lambdathe normalized Haar measure on G, L_1(G)=L_1(G,\lambda), M(G)the space of all regular Borel measures on G. We consider Banach spaces: L_1^{S^\perp}(G)=\{f\in L_1(G)\vert \int_G \gamma^{1}(g)\lambda(dg)=0 \foral \gamma\in S\}, M^{S^\perp}(G)=\{\mu\in M(G) \vert \int_G\gamma^{1}(g)\mu(dg) \forall \gamma\in S\}. Clearly L_1^{S^\perp}(G) is naturally isometric to a subspace of M^{S^\perp}(G). We investigate Banach space properties of these spaces. Answering a question of Bourgain [B] we show that L_1^{S^\perp}(G) and M^{S^\perp}(G) are not L_1spaces, although they share some properties of the classical spaces. \begin{theorem} (i) L_1^{S^\perp}(G) (resp. M^{S^\perp}(G)} is not isomorphic to any Banach lattice. (ii) There are bounded non absolutely summing operators from L_1^{S^\perp}(G) (resp M^{S^\perp}(G) ) to a Hilbert space. (iii) If E is a separable subspace of M^{S^\perp}(G) which contains L_1^{S^\perp}(G) then E is uncomplemented in M^{S^\perp}(G). \end{theorem} (iii) answers in negative a question of Plichko and Yost who asked whether so called separable complementation property (=SCP) is hereditary. Note that M^{S^\perp}(G) is a subspace of M(G) which has SCP, i.e. every separable subspace of M(G) is contained in a separable complemented subspace of M(G). \begin{theorem} (j) If G is metrizable then L_1^{S^\perp}(G) has a basis. (jj) L_1^{S^\perp}(G) has the DunfordPettis property, i.e. every weakly compact operator from L_1^{S^\perp}(G) to an arbitrary Banach space takes weakly null sequences into norm null sequences. (jjj) L_1^{S^\perp}(G) and M^{S^\perp}(G) have the uniform approximation property =(UAP). \end{theorem} Recall that a Banach space X has UAP if \exists c\ge 1 and a function \Phi:N \to N such that for every finitedimensional subspace F\subset X there is a finite rank operator T:X\to X such that T\le c, T(f)=f for f\in F \dim T(X)\le \Phi(\dim F).
I will discuss a new concept of noncommutative independence called matricial freeness, which lies somewhere in between freeness and random matrices. My talk will be similar to that in Bedlewo a few months ago, but the presentation may contain some more details.
I will discuss a new concept of noncommutative independence called matricial freeness, which lies somewhere in between freeness and random matrices. My talk will be similar to that in Bedlewo a few months ago, but the presentation may contain some more details.
We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.
We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.
We present some new classes of Generalized Brownian Motions for which the vacuum is trace state. The elementary model using random matrix was obtained by Bryc, Dembo and Jiang (Annals of Probab.) and next by another method was done by Artur Buchholz. The corresponding measure is the free product of classical Gaussian and the Wigner law or qGaussian and Wigner law. Connections with new classes of positive definite functions on Coxeter groups also will be done.
Używając metod z nieprzemiennej probabilistyki pokazujemy ze ciąg FussaCatalana \left({mp+r\atop m}\right)\frac{r}{mp+r} jest dodatnio określony o ile p\ge1, 0\le r\le p lub p\le0, p1\le r\le0. Odpowiednią miarę probabilistyczną na \mathbb{R} oznaczamy przez \mu(p,r). Pokazujemy różne własności miar \mu(p,r), na przykład zależności \begin{eqnarray*} \mu(1+p,1)^{\boxtimes t}&=&\mu(1+tp,1),\\ \mu(p_1,r)\boxtimes\mu(1+p_2,1)&=&\mu(p_1+rp_2,r),\\ \mu(a,b)\vartriangleright\mu(a+r,r)&=&\mu(a+r,b+r), \end{eqnarray*} wyliczamy też momenty potegi boolowskiej \mu(p,1)^{\uplus t}. Pokazujemy też że jeśli 0\le 2r\le p, r+1\le p lub p\le 2r+1, p\le r\le 0 to \mu(p,r) jest \boxplusnieskończenie podzielna.