All talks will be held in the Boyd Graduate Studies Center Building.
See the conference
schedule for rooms and times.
Greg Adams,
Bucknell University
Title: Analytic Reproducing Kernels
of bandwidth three and an interesting example
Abstract: This talk will focus on
reproducing kernel Hilbert spaces with orthonormal bases of the form $\{
(a_{{}_{n,0}} +a_{{}_{n,1}}z + \cdots + a_{{}_{n,J}} z^{J}) z^n , n \geq
0 \}$. The primary focus is on the tridiagonal case where $J = 1$
and how it compares to the diagonal case where $J = 0$. The
question of when multiplication by $z$ is a bounded operator is investigated
and aspects of this operator are discussed. In the well studied
diagonal case $M_z$ is a weighted unilateral shift. In the tridiagonal
case this need not be so and an example is given in which the commutant
of $M_z$ on a tridiagonal space is strikingly different from that on any
diagonal space. This is joint work with Paul McGuire.
Donald J. Bindner, Truman
State University
Title: On the space spanned by the powers of a weighted shift
and its adjoint
Classification: 47
Animikh Biswas, Universisty of North
Carolina at Charlotte
Title: Extended eigenvalues and the Volterra operator
Abstract: For a Hilbert space H, let L(H) denote the set
of all bounded linear operators on H. We say that a complex number
\lambda is an extended eigenvalue of A in L(H) if there is a nonzero X
in L(H) satisfying the equation XA=\lambda AX. The above intertwining
relation was used in order to extend the invariant subspace result of Lomonosov
by S. Brown, Kim and Pearcy, and independently by Shields. Quite
recently, some progress in this direction was made by Lauric. Here
we consider the integral Volterra operator on the space L^2(0,1).
We show that the set of extended eigenvalues of V is precisely the interval
(0,\infty) and in fact X may be chosen to be an integral operator as well.
As an applicaation of our method, it follows that the operators V and \lambda
V are not quasi-similar unless \lambda=1.
Vladimir Bolotnikov, College of
William and Mary
Title: An interpolation problem for contractive multipliers
between two reproducing kernel Hilbert spaces
Classification: 47
Abstract: Let $k_1(z,w)$ and $k_2(z,w)$ be two positive (matrix-valued)
kernels on $\Omega\times \Omega$ ($\Omega\in\C^d$) and let $H(k_1)$
and $H(k_2)$ be the corresponding reproducing kernel Hilbert spaces.
A function $S$ analytic on $\Omega$ is a contractive multiplier from $H(k_1)$
to $H(k_2)$ if the kernel $k_2(z,w)-S(z)k_1(z,w)S(w)^*$ is positive on
$\Omega\times \Omega$. A general tangential interpolation problem
will be posed for the class of such functions and the set of all solutions
characterized in terms of a certain positive kernel constructed from the
interpolation data. This positive kernel is an analogue of Potapov's fundamental
matrix inequality. It will be shown that in the special case when $\Omega$
is the unit ball of $\C^d$ and $k_1(z,w)=k_2(z,w)=(1-z_1\bar{w}_1-\ldots
-z_d\bar{w}_d)^{-1}$, the fundamental matrix inequality leads easily to
a parametrization of all solutions of the interpolation problem in terms
of a linear fractional transformation.
Rick Chartrand, The University
of Illinois at Chicago
Title: Carleson measures and multipliers for $D(\mu)$
Classification: 46
Abstract: For the Dirichlet-type space $D(\mu)$ associated
with a measure $\mu$ on the unit circle, the question of which functions
multiply the space into itself is answered in terms of
``$\mu$-Carleson measures.'' In this talk, this and other questions
and properties of $\mu$-Carleson measures will be discussed.
Raul E. Curto, The University
of Iowa
Title: Triangular Toeplitz Contractions and Cowen Sets for Analytic
Polynomials
Classification: 47
Abstract: For $f$ an analytic polynomial, let $G_{f}^{^{\prime
}}$ denote the reduced Cowen set for $f$, that is, $G_{f}^{^{\prime }}:=\{g\inH^{\infty
}(\mathbb{T}):$ $g(0)=0$ and the Toeplitz operator $T_{f+\bar{g}}$ is hyponormal$\}$.
In joint work with Muneo Cho and Woo Young Lee, we show that $G_{f}^{^{\prime
}}$ is strictly convex. We obtain this as a corollary of the following
result.
Theorem. Let $\frak{T}_{N}$ be the collection of $N\times N$ lower triangular Toeplitz contractions. Then $\frak{T}_{N}$ is compact and strictly convex in the spectral norm; that is, $\frak{T}_{N}$ is compact, convex, and $\partial \frak{T}_{N}\subseteq ext\frak{T}_{N}$.
Lower triangular Toeplitz contractions arise naturally in the solution
of the Caratheodory-Schur interpolation problem (CSIP). In fact,
CSIP is solvable iff the associated Toeplitz matrix $C$ is contractive,
i.e., $I-CC^{\ast }\geq 0$. Moreover, CSIP has a unique solution iff $\det
(I-CC^{\ast })=0$, and the unique solution is a Blaschke product. Our
proof relies on a careful analysis of the extreme points of $\frak{T}_{N}$
and its associated Blaschke products.
John Daughtry, East Carolina
University
Title: Extreme Point Questions
Abstract: We examine further questions about extreme points
in a particular Banach algebra studied by Lambert and Weinstock.
Michael Dritschel, University of
Newcastle
Title: Factorization and Model Theory
Classification: 47
Abstract: We explore the subtle connection between positive
operator-valued trigonometric polynomials and Agler's model theory.
Nathan Feldman, Washington &
Lee University
Title: Somewhere Dense Orbits are Everywhere Dense
Classification: 47A16
Abstract: We will discuss the following theorem: If the orbit
of a vector under a linear operator is somewhere dense then it must be
everywhere dense. This question was raised by Alfredo Peris and two immediate
collories are an answer to a question of Herrero's and another proof of
Ansari's Theorem, both concerning hypercyclic operators. This is joint
work with Paul Bourdon.
Lawrence Fialkow,
SUNY
at New Paltz
Title: The Truncated Complex Moment Problem: A Conjecture
Abstract: We discuss a conjectured solution to TCMP (equivalently,
to the truncated 2-dimensional real power moment problem) in terms of algebraic
and geometric invariants of the moment matrix associated to the data.
Alan Hopenwasser, University of
Alabama
Title: Regularity of Embeddings of Finite Dimensional Nest Algebras
Classification: 47L40
Abstract: I will discuss two notions of regularity, one global
and one local, for embeddings of finite dimensional nest algebras. In general,
a regular embedding is one which can be written as a sum of multiplicity
one embeddings. A locally regular embedding is one which preserves regularity
of partial isometries (with respect to some block matrix structure). Regularity
of a partial isometry, in turn, means that the block entries of the partial
isometry are again partial isometries. The main result is that, in the
context of finite dimensional nest algebras, these two notions agree. This
result is useful in the theory of direct limit algebras; in this talk,
however, I will focus on the finite dimensional linear algebra.
Zhangjian Hu, University of North
Carolina at Chapel Hill
Title: Characterizations for Some Function Spaces in the
Unit Ball of C^n
James Jamison, Univeristy of Memphis
Title: Isometric Equivalence of Some Banach Space Operators
Classification: 47A62
Abstract: Let $X$ be a Banach space and $S_1$ and $S_2$ be elements
of $B(X)$. Operators $S_1$ and $S_2$ are said to be isometrically equivalent
if there exists a surjective isometry $U$ of $X$ such that $U S_1 = S_2
U$. For example, let $X = C_p(H)$ be the Schatten class of a complex separable
Hilbert space $H$ and $A_k$ and $B_k$ ( $k =1,2$) be members of $B(H)$.
Define $S_k(T) = A_k T - T B_k.$ We give necessary and sufficient conditions
for $S_1$ to be isometrically equivalent to $S_2$. We will also give some
other results in different settings.
Srilal Krishnan, University of Alabama
Title: Principal ideals in subalgebras of groupoid C*-algebras
Classification: 47L40
Abstract: The study of different types of ideals in non
self-adjoint operator algebras has been a topic of recent research. This
paper studies one of the basic types of ideals, a principal ideal in some
non self-adjoint limit algebras. An ideal is said to be principal if it
is generated by a single element of the algebra. In this paper we look
at the ideals in subalgebras of certain groupoid C*-algebras and prove
that every ideal is principal. We obtain the same result for regular canonical
subalgebras of approximately finite C*-algebras as a particular case.
Michael Lacey, Georgia Institute
of Technology
Title: Solution of the Kato Square Root Problem
Classification: 42, 35J
Abstract: Let $L=-\hbox{div} A\nabla$ be a differential operator
in which $A$ is a bounded measurable function taking values in $d\times
d$ matricies with complex entries. Assume that ``$A$ is uniformly accreative"
in this sense. There are constants $0<\lambda<\Lambda<\infty$
for which $$\lambda|\xi|^2<\hbox{Re}\langle A(x)\xi,\xi\rangle<\Lambda|\xi|^2.$$
Then, one can define through an appropriate functional calculus a square
root of $L$. The theorem is that this operator maps the Sobolev space $H^1$
into $L^2$. That is $L^{1/2}$ acts like the gradient. This is joint work
with Pascal Auscher, Steve Hoffmann, Alan McIntosh, and Philippe Tchamitchian.
Alec Matheson, Lamar University
Title: Some remarks on regular majorants
Classification: 46
Abstract: A majorant is a continuous increasing function
$\omega(t)$ on $[0,\infty)$ with $\omega(0)=0$. The majorant $\omega(t)$
is regular if there are positive constants $c_1$ and $c_2$ such that $\int_0^t{{\omega(s)\over
s}}\,ds \le c_1\omega(t)$ and
$t \int_t^\infty{{\omega(s)\over s^2}}\,ds \le c_2\omega(t)$.
The functions $t^\alpha$,
$0< \alpha < 1$, are prototypical regular majorants. Regular
majorants are important in part because smoothness classes determined by
them are closed under harmonic conjugation. We characterize regular
majorants by showing that each is equivalent to another regular majorant
$\omega(t)$ satisfying $t^\beta\le \omega(t)\le t^\alpha$ for all $t\ge
0$ and some exponents $0<\alpha \le \beta < 1$. We give examples
to show that this characterization is best possible.
John McCarthy, Washington University
First Lecture: Pick's theorem - What's the
Big Deal ?
Abstract:
Suppose N points in the unit disk, z1, ... ,zN are
given, along with N complex numbers w1, ... , wN
. In 1916, Georg Pick considered the question of when one could find
a function f holomorphic in the unit disk and with positive
real part that interpolated the data, i.e. satisfied f(zi)
= wi for every i.
Pick completely answered the question, and his
criterion --- that the matrix whose ij'th entry is
(wi + bar(wj)) / (1 - zi
bar(zj))
be positive semi-definite --- can be proved using fairly basic function
theory.
Yet operator theorists still write hundreds of papers a year about this problem - why? The purpose of this expository talk is to explain why Pick's problem is important to engineers, and how it is related to operator theory.
John McCarthy, Washington University
Second Lecture: Generalizations of Pick's
Theorem
Abstract:
We shall discuss some extensions of Pick's
theorem to other settings.
Scott McCullough, University of
Florida
Title: Bergman-type kernels, factorization, contractive divisors,
wandering subspaces, and dilations
Classification: 47A20
Abstract: A representation theorem for a multiplier invariant
subspaces of a reproducing kernel Hilbert spaces whose kernel behaves like
the Bergman kernel is established. Consequences include contractive divisor
properties, a wandering subspace theorem, and dilation results. This is
joint work with Stefan Richter.
Ioana Mihalia, Coastal
Carolina University
Title: A Multiplicative-Periodic Function on C-{0}
Classification: 30B
Abstract: This talk will present an explicit construction
of a multiplicative-periodic meromorphic function on C-{0}. The construction
is based on considering the Riemann surfaces obtained by factoring C-{0}
through a discrete multiplicative subgroup.
Vivien Miller, Mississippi
State University
Title: Local spectral theory and weighted shifts
Classification: 47B37, 47B40
Abstract: We show that the spectrum of a decomposable
bilateral weighted shift must be a circle. As a corollary, we characterize
the weighted shifts that are generalized scalar.
Cornel Pasnicu, University
of Puerto Rico
Title: Tensor products of C*-algebras with the ideal property
Classification: 46L06, 46L05
Abstract: My talk will be based on a joint work (with
the above title) with Mikael Rordam, published in the Journal of Functional
Analysis 177,(2000),130-137. Answering into negative a conjecture
of Brown and Pedersen, Kodaka and Osaka have shown that a minimal tensor
product of real rank zero C*-algebras is not necessarily real rank zero.
We push this pathology (pertaining to what one may call noncommutative
dimension theory) much further by showing that in the case of nonexact
real rank zero C*-algebras A and B, the minimal tensor product of A with
B may not only fail to have real rank zero, but it may have ideals not
generated by projections (even in the separable case); we also point out
that this phenomenon is specific for nonexact C*-algebras. In particular,
this implies that the ideal property is not preserved under forming minimal
tensor products - even in the separable case (a C*-algebra has the ideal
property if any of its (closed,two-sided) ideals is generated (as an ideal)
by projections). The proofs rely on results by Kirchberg, Dadarlat
and myself.
Gabriel Prajitura, Bucknell University
Title: Classes of \lambda commuting operators
Classification: 47
Abstract: We consider three classes of operators on a complex
Hilbert space: operators \lambda commuting with a non-zero finite rank
operator; operators \lambda commuting with a non-zero compact operator
and operators \lambda commuting with a non-zero operator. A complete description,
in terms of the spectrum, will be given for the closures of these classes
and for some of the interiors.
Vasiliy Prokhorov, University of
South Alabama
Title: On Hankel operators associated with functions $f
\in L_p$, $ 1\le p<\infty$
Classification: 47
Abstract: We consider some questions related to the theory of
Hankel operators.
Let $G$ be a bounded simple connected domain with the boundary $\Gamma
$ consisting of a closed analytic Jordan curve. Denote by $\mm_{n,p}(G)
$, $1 \le p\le \infty$, the class of all meromorphic functions on $G$ that
can be represented in the form $h=3D\beta/\alpha ,$ where $\beta$ belongs
to the Smirnov class $ E_p(G),\ \alpha $ is a polynomial degree at most
$n ,$ $\alpha \not \equiv 0 $. We obtain estimates of
$s$-numbers of the Hankel operator $A_f $ constructed from $f \in L_p(\Gamma),
\ 1\le p<\infty,$ in terms of the best approximation
$\Delta_{n,p} $ of $f $ in the space $L_p(\Gamma) $ by functions belonging
to the class
$\mm_{n,p}(G) $.
Alexander Richman, Purdue University
Title: Spectra of composition operators in several variables
with linear fractional symbol
Abstract: Linear fractional maps on the complex plane that carry
the open unit disk into itself play a central role in developing and understanding
the theory of composition operators on the classical spaces of functions
analytic on the disk. Linear fractional maps in several variables
generalize classical linear fractional maps in the complex plane; for example,
the
class of linear fractional maps that carry the unit ball in {\bf C}$^N$
into itself include the automorphisms of the ball.
Previous work of Barbara MacCluer and Carl Cowen studied basic properties of these linear fractional maps and showed that they induce bounded composition operators on the Hardy spaces and some weighted Bergman spaces of functions analytic in the unit ball in {\bf C}$^N$. In this talk, we will review some of the earlier work on linear fractional composition operators in several variables and present new work on the spectra of these operators, focusing on the interesting case of a single fixed point on the boundary and no interior fixed points.
James Rovnyak, University
of Virginia
Title: Nudelman's problem, revisited
Classification: 47A57
Abstract: Nudelman's problem is to find Schur functions w(z)
such that b=w(A)c for given vectors b and c and an
operator A. Particular cases include classical interpolation theorems.
The main results are extended to meromorphic functions which belong to
the generalized Schur class.
Alexei Rybkin, University
of Alaska at Fairbanks
Title: On asymptotic behavior of the Titchmarsh-Weyl m-function
associated with one dimensional Schrodinger operators; some final results.
Abstract: For the general one dimensional Schrodinger
operator $-\frac{d^{2}}{dx^{2}}+q\left( x\right) $ with real $q\in L_{1}\left(\QTR{Bbb}{R}\right)
$ we present a new series representation of the Jost solution which, in
turn, implies a new asymptotic representation of the Weyl $m$-function
for locally summable $q.$ This representation is then applied to smooth
potentials $q$ to obtain Weyl $m$-function power asymptotics. We show that
the condition $q^{\left( N\right) }\in L_{1}\left( x_{0},x_{0}+\delta \right)
, N\in \QTR{Bbb}{N}_{0},$ allows one to derive the $\left( N+1\right) -$
term for almost all $x\in \lbrack x_{0},x_{0}+\delta )$ that refines relevant
results by Danielyan, Levitan and Simon. All our main results are complete.
Ilya Spitkovsky, College of William
and Mary
Title: Toeplitz Operators with Frequency Modulated Semi-Almost
Periodic Symbols
Classification: 47
Abstract: It is well known that amplitude modulation does not
affect Fredholmness of Toeplitz operators. The same is true for frequency
modulation provided the symbol of the operator is piecewise continuous.
In the present paper, it is shown that frequency modulation can destroy
Fredholmness for Toeplitz operators with almost periodic symbols; the corresponding
example is based on the observation that certain almost periodic functions
become semi-almost periodic functions after appropriate frequency modulation.
Moreover, the paper contains several results that can be employed in order
to decide whether a Toeplitz operator with a frequency modulated semi-almost
periodic symbol is Fredholm.
Tavan Trent, University of Alabama
Title: An $H^2(D^2)$-corona Theorem for Infinitely many
Functions on the Bidisk
Classification: 47A57
Abstract: We extend a corona theorem of Li (Corona Problems
of several Variables, Contemporary Mathematics,Vol.137, Amer. Math.Soc.,1991)
and Lin (M.R.94c:46106) to the case of infinitely many functions on the
bidisk.
Zhijian Wu, University of
Alabama
Title: Q spaces and Morrey spaces
Abstract: Analytic Q spaces can be described by p-Carleson measures.
By studying the derivatives of functions in Q spaces, we show that functions
in Q spaces are certain fractional differentials of functions in Morrey
spaces.
Rongwei Yang, University of Georgia
Title: Trace formula for isometric pairs
Abstract: It is well known that for every isometry $V$, $tr[V^{*},V]=-ind(V).$
This fact for the shift operator is the basis for many important developments
in operator theory as well as in topology. In this talk we show an
analogous formula for a pair of isometries $(V_{1},V_{2})$, namely $tr[V_{1}^{*},V_{1},V_{2}^{*},V_{2}]=-2ind(V_{1},V_{2})$,
where
$[V_{1}^{*},V_{1},V_{2}^{*},V_{2}]$ is the complete anti-symmetric
sum and $ind(V_{1},V_{2})$ is the Fredholm index of the pair $(V_{1},V_{2})$.
Two examples are considered.
Guoliang Yu, Vanderbilt University
First Lecture: Large scale geometry
of groups
Abstract:
Large scale geometry was first introduced
by Mostow in the proof of his famous rigidity theorem. It is popularized
by Gromov's work on geometric group theory. In this talk, I will explain
the basic ideas of large scale geometry. In particular, I will discuss
Gromov'sconcept of uniform embedding into Hilbert space.
Guoliang Yu, Vanderbilt University
Second Lecture: Novikov's Conjecture
Abstract:
A fundamental problem in the topology of high-dimensional manifolds is
the Novikov conjecture. Roughly speaking the Novikov conjecture states
that manifolds are rigid at a certain infinitesimal level. In my second
talk, I will explain what is the Novikov conjecture, why it is interesting,
and how it is related to large scale geometry of groups via K-theory of
C*-algebras.
Ruhan Zhao, University of Virginia
Title: Essential Norms of weighted composition operators
between Bloch type spaces
Classification: 47B38
Abstract: Let $D$ be the open unit disk in the complex plane.
Let $u$ be a fixed analytic function on $D$ and $\varphi$ an analytic self-map
of $D$. A weighted composition operator, which can be regarded as a generalization
of a multiplication operator and a composition operator, is defined by
$uC_{\varphi}f=uf\circ \varphi$. In this talk we estimate the essential
norms of the weighted composition operators between Bloch type spaces (including
the Lipschitz spaces $\operatorname{Lip}_\alpha$ and the Bloch space).
This is a joint work with Barbara MacCluer.