Emergent trends of Complex Analysis and Functional Analysis
(Warsaw, IM PAN)
Lecture courses II (May 29–June 8):
Monday, 29.5.2017
14.30–16.00 Nikolski I
16.00–16.45 Pause/Coffee
16.45–18.15 Nikolski II
Tuesday, 30.5.2017
14.30–16.00 Treil I
16.00–16.45 Pause/Coffee
16.45–18.15 Nikolski III
Wednesday, 31.5.2017
14.30–16.00 Nikolski IV
16.00–16.45 Pause/Coffee
16.45–18.15 Treil II
Tuesday, 6.6.2017
11.00–12.30 Seip I
14.30–16.00 Treil III
16.00–16.45 Pause/Coffee
16.45–18.15 Nikolski V
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Wednesday, 7.6.2017
11.00–12.30 Seip II
14.30–15.30 All – Polish Analysis Seminar, Mastylo I
15.30–16.15 Pause/Coffee
16.15–17.15 All – Polish Analysis Seminar, Mastylo II
17.30–19.00 Seip III
Thursday, 8.6.2017
10.00–10.55 Zarouf
11.00–12.30 Seip IV
14.30–16.00 Treil IV
16.00–16.45 Pause/Coffee
16.45–18.15 Nikolski VI
Lecture courses descriptions:
Nikolai K. Nikolski (Bordeaux University)
An introduction to the “hidden spectrum”
and norm controlled inversions
We speak on the hidden spectrum phenomenon in the case when the spectrum of certain transformations (or elements of a Banach algebra) does not
coincide with their “natural” or “visible” spectrum — for example, with the
closure of their eigenvalues. The phenomenon, first discovered for convolution measure algebras (the Wiener–Pitt phenomenon, 1938), was observed
later on for many other situations. There is a hidden spectrum effect for a
given collection of operators, or there is not, the numerical approach to that
matter is to look for an estimate of the norms of inverses (or, the resolvents)
in terms of the distance of the spectral parameter to the (visible) spectrum.
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When one deals with simultaneous inversion of a (finite) family of operators,
the problem concerns some Bezout type equations and estimates of their
solutions (“corona type” problem).
In this course, we discuss the following topics.
• Wiener–Pitt phenomenon and norm control of inverses in convolution
measure algebras on (classical) groups.
• The spectrum of Fourier multipliers on weighted L2 spaces (with and
without the hidden spectrum effect).
• Estimates of inverses for large matrices, the hidden spectrum and the
corona problem in quotient algebras H ∞ /BH ∞ .
• Hidden spectrum of certain integration operators.
No specific knowledge of the subject is supposed.
Kristian Seip (Norwegian University of Science and Technology)
Analysis for Dirichlet series and interaction with number theory
We have in recent years seen a notable growth of interest in certain functional analytic aspects of the theory of ordinary Dirichlet series
∞
X
an n−s .
n=1
Inspired by the classical theory of Hardy spaces and the operators acting on
such spaces, this topic is also intertwined with analytic number theory and
function theory on the infinite dimensional torus. Of particular interest are
problems that involve an interplay between the additive and multiplicative
structure of the integers, in this context embodied respectively by function
theory in half-planes and the so-called Bohr lift that transforms Dirichlet
series into functions of infinitely many complex variables.
During these lectures, we will discuss
• the basic function theory of Hardy spaces of Dirichlet series
• some aspects of the operator theory that has been developed for these
spaces
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• applications to the Riemann zeta function ζ(s), including pseudomoments and lower bounds for the growth of ζ(1/2 + it).
Interaction with number theory and open problems will be emphasized throughout the lectures.
Sergei Treil (Brown University)
Theory of matrix weights, revisited
Main motivations for the weighted estimates with matrix weights come
from the theory of stationary random processes, unconditional wavelet bases,
and invertibility of Toeplitz operators.
Recent developments in harmonic analysis, in particular, the technique of
domination by sparse operators and also the weighted Carleson embedding
with matrix weights, allow us to obtain new results and to simplify proofs of
the known ones.
After short introduction to the subject, I will be covering the following
topics:
• Motivations for the weighted estimates with matrix weights: stationary
processes, unconditional bases and invertibility of Toeplitz operators.
• Necessity of matrix Ap conditions for the weighted estimates of Hilbert
Transform (well known), and for the Riesz Transform (surprisingly it
was not known before).
• Technique of sparse domination of singular integral operators and the
so-called convex body valued domination as its vector-valued generalization.
• A simple proof of the weighted norm estimates with matrix weighs and
the question about the optimal order of the estimate.
• Weighted Carleson embedding with matrix weights.
• Two weight estimates with matrix weights.
• Matrix weights and perturbation theory.
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All – Polish Analysis Seminar:
Mieczyslaw Mastylo (Adam Mickiewicz University in Poznań)
Almost everywhere convergent Fourier series
In the first part of the talk we will summarize the essential known results
on almost everywhere convergent Fourier series. In the second part we will
discuss a recent work with L. Rodrı́guez-Piazza concerning the novel interpolation estimates for the Carleson maximal operators generated by partial
sums of the multiple Fourier series and all its conjugate series over cubes
defined on the d-dimensional torus. Combining these results we prove a theorem which improves all established results on the convergence of multiple
Fourier series over cubes.
Talk by Rachid Zarouf (Aix–Marseille University):
A constructive approach to Schäffer’s conjecture
We prove results that we found on our way to a deeper understanding
of Schäffer’s conjecture about inverse operators. Three topics are covered in
this work:
1. The first and main one is devoted to the question what is the smallest
S = S(n) such that
| det T | T −1 ≤ S ||T ||n−1
holds for any induced matrix norm and any invertible
complex n × n matrix.
√
J. J. Schäffer proved in 1970 that S ≤ en and conjectured that S
was bounded for any choice of invertible matrices. Schäffer’s conjecture was
rebutted by Gluskin–Meyer–Pajor using a probabilistic approach: they determined an appropriate norm and proved the existence of a sequence of
matrices T = T (n) such that S grows unboundedly. This approach was refined by J. Bourgain effectively boiling down lower estimates on S to Turántype power sum inequalities. Combined with a number theoretic analysis of
such √
inequalities H. Queffelec (1993) proved the currently strongest estimate
S ≥ n(1 − O(1/n)). However, finding the optimal S and the construction
of explicit sequences of matrices with growing S remain open tasks. Here
we provide a constructive approach to Schäffer’s conjecture. We derive new
upper estimates on S and we construct an explicit class of matrices that
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reaches the asymptotic growth due to Schäffer’s theorem. Our framework
naturally extends to provide sharp estimates on the resolvent ||(ζ − T )−1 ||
when |ζ| ≤ ||T || and ζ does not intersect the spectrum of T .
2. A key ingredient in our approach will be to investigate lp -norms of
Fourier coefficients of powers of a Blaschke factor, which is an interesting
and well-studied topic in its own right, initiated by J-P. Kahane in 1956.
3. Finally, on our way, we prove new estimates for the asymptotic behavior
of Jacobi polynomials with varying parameters and we highlight some flaws
in the established literature on this topic.
This talk is based on a joint work with Oleg Szehr.
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