# The Nikos Danikas Analysis Seminar

ARISTOTLE UNIVERSITY OF THESSALONIKI
DEPARTMENT OF MATHEMATICS
DIVISION OF MATHEMATICAL ANALYSIS
Tel. 2310 997906, Fax 2310 997945

Organizing Committee:
Ch. Konstantillaki-Savvopoulou
D.Betsakos

Seminar Room: 3rd floor in the building of the School of Sciences

# Programm of Talks (2003-2004)

This year the Seminar is dedicated to the memory of Nikos Danikas.

1. September 23, 2003
Alfonso Montes-Rodriguez,
"Highly cyclic operators, Part I"

2. September 25, 2003
Alfonso Montes-Rodriguez,
"Highly cyclic operators, Part IÉ"

3. September 30, 2003
Alesandro Rodrigeuz-Martinez,
Institute of Mathematics, Polish Academy of Sciences.
"Triangular operators and the Volterra operator".

4. October 7, 2003
Department of Physics, Aristotle University of Thessaloniki
"Calculation of operator spectra by deformed oscillator techniques"

5. October 14, 2003
Department of Physics, Aristotle University of Thessaloniki
"Calculation of operator spectra by deformed oscillator techniques"
(Part 2).

6. October 21, 2003
Th. Tselepidis,
Department of Mathematics, Aristotle University of Thessaloniki
"Estimates for the derivative of the heat kernel in hyperbolic space"

7. November 4, 2003
Th. Tselepidis,
Department of Mathematics, Aristotle University of Thessaloniki
"Estimates for the derivative of the heat kernel in hyperbolic space" (continuation).

8. November 11, 2003
M. Marias
Department of Mathematics, Aristotle University of Thessaloniki
"Strongly singular spectral multipliers on Riemannian manifolds"

9. November 25, 2003
E. Diamantopoulos
Department of Mathematics, Aristotle University of Thessaloniki
"The Hilbert matrix in Hardy spaces"

10. December 2, 2003
E. Diamantopoulos
Department of Mathematics, Aristotle University of Thessaloniki
"The Hilbert matrix in Bergman spaces"

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11. March 2, 2004
I. Gasparis
Department of Mathematics, Aristotle University of Thessaloniki
"Schauder bases in Banach spaces"
Abstract: "We shall discuss permanence properties of
Schauder basic sequences in Banach spaces (Banach's criterion
for the existence of a basis), duality of bases (biorthogonal
systems, shrinking and boundedly complete bases), special types
of bases (unconditional, symmetric, subsymmetric), examples
of bases in classical Banach spaces (The Faber-Schauder, the Haar
and the Franklin systems), the approximation property".

12. March 9, 2004
G. Styloyiannis
Department of Mathematics, Aristotle University of Thessaloniki
"Universality and entire functions: a theorem of G.Birkhoff"

13. March 16, 2004
T.L. Miller
Mississippi State University, U.S.A.
"The spectrum of generalized Cesaro operators on Hardy and weighted Bergman spaces".
Abstract: We investigate spectral properties of operators of the form
$$S_gf(z)=\frac{1}{z}\int_0^1 f(w)g(w)\,dw$$
acting on the Hardy spaces and certain weighted Bergman spaces.

14. March 23, 2004
V. Miller
Mississippi State University, U.S.A.
"Generalized Cesaro operators and compact perturbations".
Abstract: Growth conditions on the resolvent of an operator with thin spectrum are closely related to the existence of non-analytic functional calculi and thus local spectral properties of the operator. However, these properties are generally not stable under compact perturbations. Under suitable constraints however, bounds on the growth of the resolvent, and thus corresponding local spectral properties, are preserved under compact perturbations. As an application, we establish subdecomposability for certain generalized Cesaro operators on the classical Hardy spaces H^p, 1< p< \infnty.

15. March 30, 2004
I. Gasparis
Department of Mathematics, Aristotle University of Thessaloniki
"Schauder bases in Banach spaces"
Continuation of the talk of March 2, 2004.

16. April 20, 2004
G. Georganopoulos
Department of Mathematics, Aristotle University of Thessaloniki
"Properties of the Z-transform"

17. April 27, 2004
G. Georganopoulos
Department of Mathematics, Aristotle University of Thessaloniki
"Applications of the Z-transform in complex function theory and probability theory"

18. May 4, 2004
M.Marias
Department of Mathematics, Aristotle University of Thessaloniki
"Mikhlin-Hormander multipliers on graphs"

19. May 11, 2004
M.Kolountzakis
Department of Mathematics, University of Crete
"Fuglede's conjecture about orthogonal bases of exponentials"
Abstract: We present the conjecture of Fuglede that a domain $\Omega$
admits an orthogonal basis of exponential functions
$e_\lambda(x) = \exp(2\pi i \lambda x)$
if and only if it can tile space by translation. Besides showing Tao's
recent counterexample, we will discuss many partial results and argue
about the validity of the conjecture in special cases, such as convex
bodies.

20. May 12, 2004
M.Kolountzakis
Department of Mathematics, University of Crete
"Distance sets corresponding to convex bodies"
Abstract: Suppose that $K \subseteq \RR^d$, $d\ge 2$, is a $0$-symmetric convex body
which defines the usual norm
$$\Norm{x}_K = \sup\Set{t\ge 0:~~x \notin tK}$$
on $\RR^d$.
Let also $A\subseteq\RR^d$ be a measurable set of positive upper density
$\rho$.
We show that if the body $K$ is not a polytope, or if it is a polytope
with
many faces (depending on $\rho$),
then the distance set
$$D_K(A) = \Set{\Norm{x-y}_K:~~x,y\in A}$$
contains all points $t\ge t_0$ for some positive number $t_0$.
This was proved by Furstenberg, Katznelson and Weiss,
by Falconer and Marstrand and by Bourgain in the case where $K$ is
the Euclidean ball in any dimension greater than $1$.
As corollaries we obtain (a) an extension to any dimension
of a theorem of Iosevich and \L aba
regarding distance sets with respect to convex bodies of well-distributed
sets in the plane,
and also (b) a new proof of a theorem of Iosevich, Katz and Tao about the
nonexistence of Fourier spectra for smooth convex bodies with positive
curvature.

21. May 18, 2004
M. Anousis
Department of Mathematics, University of the Aegean
Title: "Geometric compact elements in C* algebras".
Abstract: We consider a C* algebra A and an element a in A. We give
a necessary and sufficient (geometric) condition
for the existence of a representation ö of A such that
ö(a) is a compact operator.
We also give a caracterization of the compact adjointable
operators on a countably generated Hilbert module.

22. May 25, 2004
D. Betsakos
Department of Mathematics, Aristotle University of Thessaloniki
Title: "An inverse problem in Potential Theory"
Abstract: We say that a logarithmic potential generates a curve in the plane if a unit mass traces the curve under the action of the potential. We consider the following problem: A one-parameter family of plane curves is given. We assume that these curves lie in the complement of a compact set $K$. Find all measures supported in $K$ that generate each of the given curves. We solve this problem when $K$ is the unit circle in three specific cases: (a) when the given curves are straight lines through the origin; (b) when the curves are straight lines through a point on the unit circle, and (c) when the curves are circles centered at the origin. The solution involves the Poisson integral and its boundary behavior.

The Analysis Seminar in 2002-2003.(in Greek)
The Analysis Seminar in 1992-2002. (in Greek)