Analysis Seminar
Unless specified, all seminars are
Wednesday 4-5pm at 250 Math Building.
August
27th
Michio Seto, Shimane University,
Japan
A
Rank Problem on Submodules in the Hardy Space over the
Bidisk
September 11th
Zhong-Jin Ruan, University of Illinois at
Urbana-Champaign
(Thursday,
Colloquium) Operator
Spaces and Their Applications to Abstract Harmonic
Analysis
Abstract: The theory of operator spaces is a natural
(noncommutative)
quantization
of
Banach space theory. It has already had many important
applications to
operator
algebras, abstract harmonic analysis and some other related areas.
In this
talk, I will
first review some basic notions in operator spaces. Then I will
show a
representation
theorem related
to the measure algebra M(G) and the completely bounded
Herz-Schur
multiplier algebra McbA(G) on the operator space
B(L2(G)), where G
is a
locally
compact
group.
September
17th
Jingbo Xia, SUNY at
Buffalo
One
Hundred Years of Diagonalization, Part
I
Abstract: In
1909, Hermann Weyl published a theorem which, stated in modern
terms,
says
that every self-adjoint operator A on a separable Hilbert space H can
be
diagonalized
modulo
a compact operator. Since then this result has been improved and
generalized
in
many different ways. Most noticeably, improvements and generalizations
have
come
in
the direction of the simultaneous diagonalization of commuting tuples (A1,
...,
An)
of
self-adjoint operators modulo norm ideals. Modern-day diagonalization
requires
techniques
and ideas from harmonic analysis, such as dyadic decomposition,
stopping-time
argument,
singular integral operators, etc. In these talks we will review the
progress
made
in
the hundred years since Weyl's paper, culminating in the latest result.
September
24th Jingbo
Xia, SUNY at
Buffalo
One
Hundred Years of Diagonalization, Part II
October
1st Jingbo
Xia, SUNY at Buffalo
One Hundred Years of Diagonalization, Part III
October
8th
Anthony Weston, Canisius College
Determining lower bounds on the maximal p-negative type of finite
metric spaces
Abstract:
The metric notions of negative type (1910s) and generalized roundness (1960s)
were introduced to study isometric
and uniform embeddings, respectively. More recently,
these notions (which are actually equivalent) have found serious
applications in coarse
geometry and combinatorial
optimization.
The purpose of this talk will be to present a simple (but alarmingly effective)
method for determining lower bounds
on the maximal p-negative type of finite metric spaces.
These bounds only depend upon the cardinality and the (scaled) diameter of the
underlying
metric space. Moreover,
these bounds can easily be optimal, as we shall explain with
examples towards the end of the talk. I will also examine the special case
of finite metric
trees in some detail.
October
15th
Joshua Isralowitz, SUNY at Buffalo
Toeplitz operators with BMO symbols on the Segal-Bargmann space
Abstract: We show that
Zorboska's criterion for compactness of Toeplitz operators with
BMO1 symbols on the Bergman space of the unit disc holds, by
a
different proof, for the
Segal-Bargmann space of Gaussian square-integrable entire functions on Cn.
We establish
some basic properties of BMOp for p≥1
and complete
the characterization of bounded
and compact Toeplitz operators with BMO1
symbols. Via
the Bargmann isometry and
results of Lo and Engliš, we
also
give a compactness
criterion for the Gabor-Daubechies
"windowed Fourier localization operators" on
L2(Rn,
dv) when the symbol is in a BMO1
Sobolev-type space. Finally, we
discuss
examples of the compactness criterion and
counter-examples to the unrestricted
application of this criterion for the compactness of
Toeplitz operators. This is joint work with
Lewis Coburn and Bo Li.
October
22nd
Ruhan Zhao, SUNY at Brockport
An Excursion to Qp Spaces
Abstract: The Qp
(0<p<∞) spaces
were introduced by Aulaskari, Xiao and Zhao
in 1995.
It generalizes the classical space BMOA and the Bloch space B in the
sense that Q1=BMOA
and Qp=B for any p>1. For 0<p<1, Qp
spaces are
subspaces of BMOA. Since it was
introduced in 1995, many authors have studied these spaces
and found connections with
many other classical function spaces. In this talk we will take a
tour on these Qp spaces.
We will focus on some basic properties and some recent development on
these spaces.
We will also look at the generalizations of Qp spaces to
analytic
functions
on unit ball of
Cn and to real
valued functions on Rn.
October
29th
Masoud Khalkhali, University of Western
Ontario
The Algebra of Formal Twisted
Pseudodifferential Symbols and a Noncommutative Residue
Abstract: We extend the
Adler-Manin trace on the algebra of
pseudodifferential symbols to
a twisted setting.
November
5th
Tao Mei, University of
Illinois at Urbana-Champaign
Riesz Transforms in the Noncommutative
Setting
Abstract: In this talk I
will review the boundedness of Riesz transforms in the classical
setting
and P. A. Meyer's
formulation of Riesz tranforms by semigroups of operators. Then I will
introduce a recent work (joint with M. Junge) on noncommutative Riesz
transforms and
applications to quantum metic spaces.
November
12th
Pinhas Grossman, Vanderbilt University
Quadrilaterals of Factors
Abstract:
A quadrilateral of factors is a pair of intermediate subfactors P, Q of an
irreducible finite-index inclusion of II1 factors N⊂M
such that P and Q
generate M
and intersect in N. For certain noncommuting
quadrilaterals,
there is a rigidity to the inclusions
which imposes severe
restrictions on invariants such indices and angles. This is joint work
with Masaki Izumi.
Some of these quadrilaterals can also be used to estimate the strong-singularity
constants of Sinclair and Smith for subfactors. In particular, there is a
subfactor which is singular
but not strongly singular.
This is joint work with Alan Wiggins.
November
19th
Quanlei Fang, SUNY at Buffalo
Explicit transfer function realization for
two-variable rational functions
via zero/pole data
Abstract: A scalar-valued
rational function of two complex variables, as the ratio of two
polynomials which are relatively prime, is determined by its
pole curve and zero curve up to
multiplicity. A transfer-function realization for the
function (of Givone-Roesser
or
Fornasini-Marchesini type), if minimal in the Popov-Belevitch-Hautus
sense, determines
a linearization (i.e., a determinantal representation) for the pole
curve and the zero curve.
We discuss the converse
question of constructing a realization (of either Givone-Roesser or
Fornasini-Marchesini type) for a function having prescribed zero and pole
curvess.
The basic idea follows the solution for the one-variable case due to
Ball-Gohberg-Rodman,
but with additional ingredients from the theory of determinantal
representations for
algebraic
curves. This is a joint work with Joseph Ball and Victor Vinnikov.