###
11/11/2014, 16:30 — 17:30 — Room P4.35, Mathematics Building

Piotr Budzyński, *University of Agriculture in Krakow*

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Invitation to weighted shifts on directed trees

Weighted shifts on directed trees form an important class of operators introduced recently in [6]. This class is a natural and substantial generalization of the class of classical (unilateral or bilateral) weighted shifts on $\ell^2$ spaces. It is also related to a class of composition operators in $L^2$-spaces.

Weighted shifts on directed trees have proven to have very interesting features (see [2, 3, 5, 6, 7]). The underlying relatively simple graph structure gives a rise to a subtle and complex structure of the operators, which turn out to have properties not known before in other classes of operators, and makes them ideal for testing hypothesises and constructing examples. We will outline recent results concerning these operators with main emphasis on on subnormality and reflexivity.

The talk is based on a joint work with Z.J. Jabloński, I.B. Jung and J. Stochel, and M. Ptak.

### References

- P. Budzyński, P. Dymek, Z. J. Jabłoski, J. Stochel, Subnormal weighted shifts on directed trees and composition operators in $L^2$-spaces with non-densely defined powers,
*Abstract Appl. Anal.* (2014), Article ID 791817, 6 pages. - P. Budzyński, Z. J. Jabłoski, I. B. Jung, J. Stochel, Subnormality of unbounded weighted shifts on directed trees,
*J. Math. Anal. Appl.* 394 (2012), 819-834. - P. Budzyński, Z. J. Jabłoski, I. B. Jung, J. Stochel, Subnormality of unbounded weighted shifts on directed trees. II,
*J. Math. Anal. Appl.* 398 (2013), 600-608. - P. Budzyński, Z. J. Jabłoski, I. B. Jung, J. Stochel, Unbounded subnormal composition operators in $L^2$-spaces,
*preprint*, http://arxiv.org/abs/1310.3542. - P. Budzyński, Z. J. Jabłoski, I. B. Jung, J. Stochel, Subnormal weighted shifts on directed trees whose nth powers have trivial domain,
*preprint*, http://arxiv.org/abs/1409.8022. - Z. J. Jabłoński, I. B. Jung, J. Stochel, Weighted shifts on directed trees,
*Mem. Amer. Math. Soc* 216 (2012), no. 1017. - Z. J. Jabłoński, I. B. Jung, J. Stochel, A non-hyponormal operator generating Stieltjes moment sequences,
*J. Funct. Anal.* 262 (2012), 3946-3980.

###
07/11/2014, 11:00 — 12:00 — Room P4.35, Mathematics Building

Abdelhamid Boussejra, *University Ibn Tofail, Kenitra, Marocco*

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Poisson integrals on Riemannian Symmetric Spaces

In this talk we shall give characterizations of the $L^{p}$-range of the Poisson transform $P_{\lambda}$ associated to Riemannian Symmetric Spaces. We will focus on the rank one symmetric space case, and show that for $\lambda$ real, the Poisson transform is a bijection from the space of $L^{2}$ functions on the boundary (respectively $L^{p}$) onto a subspace of eigenfunctions of the Laplacian satisfying certain $L^{2}$-type norms (respectively Hardy-type norms).

###
28/10/2014, 16:30 — 17:30 — Room P4.35, Mathematics Building

Sergey Naboko, *The University of Kent*

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Spectral analysis of Jacobi operators generated by Markov Birth and Death Processes

Some particular examples of Jacobi Operators (tridiagonal matrices) with growing entries related to the Markov processes will be considered.

Using a Levinson's type theorems approach we plan to determine the spectral structure of the corresponding operators.

No preliminary knowledge of Jacobi Matrices or Orthogonal Polynomials to be required.

###
07/10/2014, 14:00 — 15:00 — Room P4.35, Mathematics Building

Gwion Evans, *Aberystwyth University*

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Characterising Higher-Rank Graph C*-Algebras

There is an elegant theory for graph C*-algebras that allows one to determine structural properties of the C*-algebra from the underlying directed graph. By coupling this with C*-algebra classification results one can characterise many graph C*-algebras as falling into various known classes of nuclear classifiable C*-algebras. Whereas much of the structural theory carries over, the C*-algebras associated to higher-rank analogues of directed graphs are much less well-understood. I will recall the standard tools that are available to study higher-rank graph C*-algebras and discuss how recent developments in Elliot's classification programme could be used to help characterise higher-rank graph C*-algebras.

###
06/05/2014, 16:30 — 17:30 — Room P3.10, Mathematics Building

Karim Kellay, *Université Bordeaux I, France*

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Sampling, interpolation and Riesz bases in the small Fock spaces.

We give a complete description of Riesz bases and characterize
interpolation and sampling in terms of densities.

This is joint work with A. Baranov, A. Dumont and A.
Hartmann.

###
28/04/2014, 16:30 — 17:30 — Room P3.10, Mathematics Building

Alexandre Almeida, *Universidade de Aveiro*

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Integral operators and elliptic equations in variable exponent
Lebesgue spaces.

We study mapping properties of variable order Riesz and Wolff
potentials within the framework of variable exponent Lebesgue
spaces. As an application, optimal integrability results for
solutions to the \(p(.)\)-Laplace equation are given in the scale
of (weak) Lebesgue spaces.

###
13/03/2014, 16:30 — 17:30 — Room P3.10, Mathematics Building

David Krejcirik, *Nuclear Physics Institute ASCR, Czech Republic*

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Pseudospectra in non-Hermitian quantum mechanics

We propose giving the mathematical concept of the pseudospectrum a
central role in quantum mechanics with non-self-adjoint operators.
We relate pseudospectral properties to quasi-self-adjointness,
similarity to self-adjoint operators and basis properties of
eigenfunctions. Applying microlocal techniques for the location of
the pseudospectrum of semiclassical operators to models familiar
from recent physical literature, unexpected wild properties of the
operators are revealed. This is joint work with Petr Siegl, Milos
Tater and Joe Viola.

###
13/02/2014, 16:30 — 17:30 — Room P3.10, Mathematics Building

Cristina Câmara, *Instituto Superior Técnico*

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Riemann-Hilbert problems, Toeplitz operators and \(Q\)-classes

We generalize the notion of $Q$-classes ${C}_{{Q}_{1},{Q}_{2}}$, which was
introduced in the context of Wiener-Hopf factorization, by
considering very general $2\times 2$ matrix functions ${Q}_{1}$, ${Q}_{2}$.
This allows us to use a mainly algebraic approach to obtain several
equivalent representations for each class, to study the
intersections of $Q$-classes and to explore their close connection
with certain non-linear scalar equations. The results are applied
to various factorization problems and to the study of Toeplitz
operators with symbol in a $Q$-class.

###
04/12/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building

Antti Perälä, *University of Helsinki, Finland*

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Optimal bounds for analytic projections

We discuss some recent advances related to size estimates of
analytic projections and the possible uses for such estimates in
applications. The spaces considered include Hardy, Bergman, Bloch,
Besov and Segal-Bargmann spaces. We study in detail the case of
Bergman projection onto the maximal and minimal Möbius invariant
spaces.

###
14/11/2013, 14:00 — 15:00 — Room P3.10, Mathematics Building

Abdelhamid Boussejra, *Université Ibn Tofail, Kenitra, Morocco.*

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The Hua operators on homogeneous line bundles over bounded
symmetric domains of tube type

Let $\mathcal{D}=G/K$ be a bounded symmetric domain of tube
type. We show that the image of the Poisson transform on the
degenerate principal series representation of $G$ attached to the
Shilov boundary of $\mathcal{D}$ is characterized by a $K$-
covariant differential operator on a homogeneous line bundle over
$\mathcal{D}$. As a consequence of our result we get the
eigenvalues of the Casimir operator for Poisson transforms on
homogeneous line bundles over $G/K$. This extends a result of
Imemura and all on symmetric domains of classical type to all
symmetric domains. Also we compute a class of Hua type integrals
generalizing an earlier result of Faraut and Koranyi.

###
06/11/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building

Sergey Naboko, *University of Kent and St.Petersburg State University*

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Spectral analysis of Jacobi Matrices and asymptotic properties of
orthogonal polynomials

We review basic features of the spectral theory of Hermitian Jacobi
operators. The analysis is based on asymptotic properties of the
related orthogonal polynomials at infinity for fixed spectral
parameter. We discuss various examples of bounded and unbounded
Jacobi matrices. This talk is meant to give an introduction to the
theory of Jacobi matrices and orthogonal polynomials.

###
18/07/2013, 18:00 — 19:00 — Room P3.10, Mathematics Building

Carl Cowen, *Indiana University-Purdue University Indianapolis, USA*

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Rota's Universal Operators and Invariant Subspaces in Hilbert
Spaces

Rota showed, in 1960, that there are operators $T$ that provide
models for every bounded linear operator on a separable, infinite
dimensional Hilbert space, in the sense that given an operator $A$
on such a Hilbert space, there is $\lambda \ne 0$ and an invariant
subspace $M$ for $T$ such that the restriction of $T$ to $M$ is
similar to $\lambda A$. In 1969, Caradus provided a practical
condition for identifying such universal operators. In this talk,
we will use the Caradus theorem to exhibit a new example of a
universal operator and show how it can be used to provide
information about invariant subspaces for Hilbert space operators.
Of course, Toeplitz operators and composition operators on the
Hardy space ${H}^{2}(\mathbb{D})$ will play a role!

This talk describes work in collaboration with Eva
Gallardo-Gutiérrez, Universidad Complutense de Madrid, done there
this year during the speaker's sabbatical.

###
18/07/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

David Krejcirik, *Nuclear Physics Institute ASCR, Czech Republic*

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The Brownian traveller on manifolds

We study the inﬂuence of the intrinsic curvature on the large
time behaviour of the heat equation in a tubular neighbourhood of
an unbounded geodesic in a two-dimensional Riemannian manifold.
Since we consider killing boundary conditions, there is always an
exponential-type decay for the heat semigroup. We show that this
exponential-type decay is slower for positively curved manifolds
comparing to the ﬂat case. As the main result, we establish a
sharp extra polynomial-type decay for the heat semigroup on
negatively curved manifolds comparing to the ﬂat case. The proof
employs the existence of Hardy-type inequalities for the Dirichlet
Laplacian in the tubular neighbourhoods on negatively curved
manifolds and the method of self-similar variables and weighted
Sobolev spaces for the heat equation.

- Martin Kolb and David Krejcirik: The Brownian traveller on
manifolds, J. Spectr. Theory, to appear; preprint on arXiv:1108.3191
[math.AP].

###
20/06/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

Nelson Faustino, *Universidade de Coimbra*

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Berezin Calculus over Weighted Bergman Spaces of Polyanalytic type

Starting from the Poincaré metric $d{s}^{2}=\frac{1}{2\pi i}{(1-\mid z{\mid}^{2})}^{-2}d\stackrel{\u203e}{z}\phantom{\rule{thickmathspace}{0ex}}dz$ on the the unit
disk $\mathbb{D}$, we will study the range of the Berezin
transforms generated from the normalized kernel function
${K}_{\zeta}^{n}(z)={K}^{n}(z,\zeta ){K}^{n}(\zeta ,\zeta {)}^{-\frac{1}{2}}$
regarding the weighted polyanalytic Bergman spaces
${A}_{n}^{\alpha}(\mathbb{D})$ of order $n$. Special emphasize will be
given to the invariance of the range of the Berezin transformation
under the action of the Möbius transformations
${\phi}_{\zeta}(z)=\frac{z-a}{1-\stackrel{\u203e}{\zeta}z}$. Connection
between Berezin calculus over weighted Bergman spaces of
polyanalytic type on the disk $\mathbb{D}$ and on the upper half
space ${\u2102}^{+}$ will also be discussed along the talk.

###
16/05/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

Sérgio Mendes, *Instituto Universitário de Lisboa, ISCTE-IUL*

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Noncommutative summands of the ${C}^{*}$-algebra ${C}_{r}^{*}{\mathrm{SL}}_{2}({\mathbb{F}}_{2}((\varpi )))$

Let ${\mathbb{F}}_{2}((\varpi ))$ denote the Laurent series in the
indeterminate $\varpi $ with coefficients over the finite field with
two elements ${\mathbb{F}}_{2}$. This is a local nonarchimedean field
with characteristic $2$. We show that the structure of the reduced
group ${C}^{*}$-algebra of ${\mathrm{SL}}_{2}({\mathbb{F}}_{2}((\varpi )))$ is determined
by the arithmetic of the ground field. Specifically, the algebra
${C}_{r}^{*}{\mathrm{SL}}_{2}({\mathbb{F}}_{2}((\varpi )))$ has countably many
noncommutative summands, induced by the Artin-Schreier symbol. Each
noncommutative summand has a rather simple description: it is the
crossed product of a commutative ${C}^{*}$-algebra by a finite group.
The talk will be elementary, starting from the scratch with the
definition of ${C}_{r}^{*}{\mathrm{SL}}_{2}$.

###
18/04/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

Gabriel Cardoso, *Instituto Superior Técnico*

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A light introduction to supersymmetry

We give a brief introduction to supersymmetric quantum mechanics.

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21/03/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

Cristina Câmara, *Instituto Superior Técnico*

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A Riemann-Hilbert approach to Toeplitz operators and the corona
theorem

Together with differential operators, Toeplitz operators (TO)
constitute one of the most important classes of non-self adjoint
operators , and they appear in connection with various problems in
physics and engineering. The main topic of my presentation will be
the interplay between TOs and Riemann-Hilbert problems (RHP), and
the relations of both with the corona theorem. It has been shown
that the existence of a solution to a RHP with $2\times 2$
coefficient $G$, satisfying some corona type condition, implies –
and in some cases is equivalent to – Fredholmness of the TO with
symbol $G$. Moreover, explicit formulas for an appropriate
factorization of $G$ were obtained, allowing to solve different
RHPs with coefficient $G$, and to determine the inverse, or a
generalized inverse, of the TO with symbol $G$. However, those
formulas depend on the solutions to 2 meromorphic corona problems.
These solutions being unknown or rather complicated in general, the
question whether the factorization of $G$ can be obtained without
the corona solutions is a pertinent one. In some cases, it already
has a positive answer; how to solve this question in general is
open, and all the more so in the case of $n\times n$ matrix
functions $G$, for which the results regarding the $2\times 2$ case
have recently been generalized.

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18/02/2013, 14:00 — 15:00 — Room P3.10, Mathematics Building

Pedro Patrício, *Universidade do Minho*

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Generalized invertibility in rings: some recent results

The theory of generalized inverses has its roots both on semigroup
theory and on matrix and operator theory. In this seminar we will
focus on the study of the generalized inverse of von Neumann,
group, Drazin and Moore-Penrose in a purely algebraic setting. We
will present some recent results dealing with the generalized
inverse of certain types of matrices over rings, emphasizing the
proof techniques used.

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17/01/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

Petr Siegl, *Universidade de Lisboa*

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Spectral analysis of some non-self-adjoint operators

We give an introduction to the study of one particular class of
non-self-adjoint operators, namely
$\mathcal{P}\mathcal{T}$-symmetric ones. We explain briefly the
physical motivation and describe the classes of operators that are
considered. We explain relations between the operator classes,
namely their non-equivalence, and mention open problems.

In the second part, we focus on the similarity to self-adjoint
operators. On the positive side, we present results on
one-dimensional Schrödinger-type operators in a bounded interval
with non-self-adjoint Robin-type boundary conditions. Using
functional calculus, closed formulas for the similarity
transformation and the similar self-adjoint operator are derived in
particular cases. On the other hand, we analyse the imaginary cubic
oscillator, which, although being
$\mathcal{P}\mathcal{T}$-symmetric and possessing real spectrum, is
not similar to any self-adjoint operator. The argument is based on
known semiclassical results.

- P. Siegl: The non-equivalence of pseudo-Hermiticity and
presence of antilinear symmetry, PRAMANA-Journal of Physics, Vol.
73, No. 2, 279-287,
- D. Krejcirík, P. Siegl and J. Zelezný: On the similarity of
Sturm-Liouville operators with non-Hermitian boundary conditions to
self-adjoint and normal operators, Complex Analysis and Operator
Theory, to appear,
- P. Siegl and D. Krejcirík: On the metric operator for
imaginary cubic oscillator, Physical Review D, to appear.

###
06/12/2012, 16:30 — 17:30 — Room P4.35, Mathematics Building

Cristina Diogo, *ISCTE-IUL and CAMGSD*

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Corona conditions and symbols with a gap around zero

Convolution equations on a finite interval (which we can assume to
be $[\mathrm{0,1}]$) lead to the problem of factorizing matrix functions $$G=\left[\begin{array}{cc}{e}_{-1}& 0\\ g& {e}_{1}\end{array}\right]$$
where ${e}_{\theta}(\xi )={e}^{i\theta \xi}$, $\theta \in \mathbb{R}$ and
$g\in {L}_{\mathrm{\infty}}(\mathbb{R})$. Here we consider $g$ of the form
$$g={a}_{+}{e}_{\mu}+{a}_{-}{e}_{-\sigma}$$ with ${a}_{\pm}\in {H}_{\mathrm{\infty}}({\u2102}^{\pm})$ and $\mu ,\sigma >0$. Imposing some
corona-type conditions on ${a}_{\pm}$, we show that solutions to the
Riemann-Hilbert problem $G{h}_{+}={h}_{-}$, with ${h}_{\pm}\in ({H}_{\mathrm{\infty}}({\u2102}^{\pm}){)}^{2}$, can be determined explicitly and conditions
for invertibility of the Toeplitz operator with symbol $G$ in
$({H}_{p}^{+}{)}^{2}$ can be derived from them.