Operator Theory Seminar

The Operator Theory Seminar at IUPUI has met regularly since January 2007.

This page is a record, in reverse chronological order, of the topics and speakers that have been included.

SPRING 2009

End of Spring Semester 2009

Fall 2008

Spring 2008

Fall 2007

Summer 2007

Spring 2007

Beginning of Seminar

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April 1

Carl Cowen (IUPUI)

Stephan Garcia's Work on Complex Symmetric Operators (continued)

March 25

Carl Cowen (IUPUI)

Stephan Garcia's Work on Complex Symmetric Operators

March 18

No Meeting (IUPUI Spring Break)

March 11     (Seminar will start at 3:00p today)

Michael Jury (University of Florida)

Valiron's Theorem on the Unit Ball

Abstract: Valiron's theorem in the disk says that every hyperbolic self-map of the disk is conjugate (by a not necessarily invertible transformation) to a hyperbolic automorphism. I will describe a weak version of this theorem in the unit ball of C^n, under a mild smoothness assumption on the map, and a related result of Bracci, Gentili, and Poggi-Corradini. As an application, we obtain a formula for the spectral radii of hyperbolic Schur class composition operators.

March 4

Carl Cowen (IUPUI)

Integration on spheres

February 25

NO MEETING (IUPUI)

February 18

Carl Cowen (IUPUI)

Integration on spheres

February 11

Carl Cowen (IUPUI)

Integration on spheres

February 4

Eva Gallardo-Gutierrez (University of Zaragoza, Spain)

replaced by Carl Cowen (IUPUI)

Discussion on "On the path component of compact composition operators in the Hardy space (conclusion)"

January 28 Campus Closed; seminar postponed as above

January 28

Eva Gallardo-Gutierrez (University of Zaragoza, Spain)

On the path component of compact composition operators in the Hardy space (conclusion)

January 21

Eva Gallardo-Gutierrez (University of Zaragoza, Spain)

On the path component of compact composition operators in the Hardy space

January 14 (First meeting of the semester)

Carl Cowen (IUPUI)

Integrals related to H^2(B^N)

FALL 2008

End of Fall Semester 2008

Spring 2009

Spring 2008

Fall 2007

Summer 2007

Spring 2007

Beginning of Seminar

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December 3 (Last meeting of the semester)

William Higdon (University of Indianapolis)

On The Numerical Range Of A Class Of Composition Operators on H^2

November 26

No Meeting -- Thanksgiving Break

November 19

Discussion on problems in Operator Theory

November 12

Morteza Seddighin (IU East)

Anti-eigenvalues and Operator Inequalities

Abstract

The Kantrovich inequality and Kantrovich-like inequalities for positive bounded operators have had many applications in operator theory, and other areas since 1948. The concept of Antieigenvalue or the operator cosine was independently introduced in 1967 by Karl Gustafson for use in perturbation of semigroup generators. For a positive operator, both the Kantorovich inequality and the value of operator cosine are expressed in terms to the smallest and largest eigenvalues of the operator, and it turns out that the former is obtained from the latter. We will discuss the extensions of Kantorovich inequality and Antieigenvalue to other classes of operators.

November 5

Carl Cowen (IUPUI)

Maps of the unit ball in the Schur-Agler Class (continued)

October 29

Carl Cowen (IUPUI)

Maps of the unit ball in the Schur-Agler Class (continued)

October 22

Carl Cowen (IUPUI)

Maps of the unit ball in the Schur-Agler Class (continued)

October 15

Carl Cowen (IUPUI)

Maps of the unit ball in the Schur-Agler Class (continued)

October 8

Carl Cowen (IUPUI)

Maps of the unit ball in the Schur-Agler Class (continued)

October 1

Carl Cowen (IUPUI)

Maps of the unit ball in the Schur-Agler Class

September 24

Carl Cowen (IUPUI)

A class of analytic maps of the unit ball in C^N into itself

September 17

Carl Cowen (IUPUI)

Hermitian Weighted Composition Operators on Weighted Hardy Spaces (continued)

September 10

Carl Cowen (IUPUI)

Hermitian Weighted Composition Operators on Weighted Hardy Spaces (continued)

September 3

Carl Cowen (IUPUI)

Hermitian Weighted Composition Operators on Weighted Hardy Spaces

This talk describes work of Gajath Gunatillake, Eungil Ko, and the speaker that extends work in an earlier paper that describes the operators of the title acting on the usual Hardy Hilbert space. A 'Weighted Hardy Space' is a Hilbert space of functions analytic on the unit disk in which the polynomials are dense and the monomials are an orthogonal set. The usual Hardy space, the Dirichlet space, the Bergman space, and weighted Bergman spaces with a radially symmetric wieght function are all examples of a weighted Hardy space.

Necessary conditions for weighted composition operator are given for arbitrary weighted Hardy spaces and these are used to identify all the Hermitian weighted composition operators on the standard Bergman space. In addition, the results on weighted composition operators are used to find the extremal functions, related to contractive divisors, for some specific invariant subspaces for multiplication by z on the Bergman space.

SPRING 2008

End of Spring Semester 2008

Spring 2009

Fall 2008

Fall 2007

Summer 2007

Spring 2007

Beginning of Seminar

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April 23

Carl Cowen (IUPUI)

Linear Fractional Maps in C^N and their Composition Operators (continued)

April 16

Carl Cowen (IUPUI)

Linear Fractional Maps in C^N and their Composition Operators (continued)

April 9

Carl Cowen (IUPUI)

Linear Fractional Maps in C^N and their Composition Operators (continued)

April 2

Carl Cowen (IUPUI)

Linear Fractional Maps in C^N and their Composition Operators (continued)

March 26

Carl Cowen (IUPUI)

Linear Fractional Maps in C^N and their Composition Operators (continued)

March 19

Carl Cowen (IUPUI)

Linear Fractional Maps in C^N and their Composition Operators

March 12

NO MEETING -- Spring Break!

March 5

Discussion

February 27

Discussion

February 20

William Higdon (University of Indianapolis)

Composition Operators Induced by Parabolic Linear Fractional Self-Maps of the Disk (continued)

February 13

William Higdon (University of Indianapolis)

Composition Operators Induced by Parabolic Linear Fractional Self-Maps of the Disk

February 6

Carl Cowen (IUPUI)

Invariant subspaces for multiplication operators (concluded)

January 30

Carl Cowen (IUPUI)

Invariant subspaces for multiplication operators (continued)

January 23

Carl Cowen (IUPUI)

Invariant subspaces for multiplication operators (continued)

January 16

Carl Cowen (IUPUI)

Invariant subspaces for multiplication operators

There are several famous theorems on the invariant subspaces of a multiplication operator, especially those of Beurling and the results on the operator of multiplication by z on the Bergman space. The subject begins with the invariant subspaces of a diagonal operator (i.e. a normal operator), but quickly becomes quite complicated and intricate. The goal of this series of talks will be to summarize, with few proofs, a recent long paper by Aleman, Feldman, and Ross on the invariant subspaces of the operator of multiplication (on the Hardy space) by the Riemann map of the disk onto a slit disk. The talks will begin with the ideas from the simpler settings first.

FALL 2007

End of Fall Semester 2007

Spring 2009

Fall 2008

Spring 2008

Summer 2007

Spring 2007

Beginning of Seminar

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December 5

Conversation about Numerical Range and Composition Operators

Final Meeting of Fall Semester

November 28

Conversation about Numerical Range and Composition Operators

November 21

No Meeting: Thanksgiving Recess

November 14

Rebecca Wahl (Butler Univ.)

Invariant Subspaces for a Class of Composition Operators

November 7

Bill Higdon (Univ. of Indianapolis)

Toeplitz-Hausdorff Theorem: the Numerical Range Is Convex

October 31

Bill Higdon (Univ. of Indianapolis)

J. H. Shapiro's Notes on the Numerical Range (continued)

October 24

Carl Cowen (IUPUI)

Computing the Numerical Range

October 17

Bill Higdon (Univ. of Indianapolis)

J. H. Shapiro's Notes on the Numerical Range

For an n x n matrix A, the numerical range of A is the set of numbers in the complex plane < Ax, x > for x a vector of length 1 in complex n-space. A surprising theorem of Toeplitz from the late 19th century asserts this is a convex set. It seems that the numerical range should be a useful tool in the study of operators on a Hilbert space, but up to this time, it has not been much exploited. This will be an introduction to the subject.

October 10

Continuation of Discussion on uses of Halmos's Problem on Multiplication Operators

October 3

Continuation of Discussion on uses of Halmos's Problem on Multiplication Operators

September 26

Continuation of Discussion on uses of Halmos's Problem on Multiplication Operators

September 19

Discussion on uses of Halmos's Problem on Multiplication Operators

Halmos's book, "A Hilbert Space Problem Book", gives a characterization of multiplication operators on functional Hilbert spaces. In his talk of September 12, Professor Montes-Rodriguez used this characterization in a key step to construct an isomorphism between H^2(D) and the Sobolev space, W^{1,2}([0,\infty)). Recently other researchers have used similar constructions to good effect. This meeting of the seminar will be a discussion by participants on the characterization and its recent uses in the study of composition operators.

September 12

Prof. Alfonso Montes-Rodriguez (Univ. of Seville, Spain)

The Lattice of Parabolic Non-automorphism Composition Operators on the Hardy Space

The lattice of composition operators induced by parabolic non-automorphisms in the Hardy space will be characterized. The knowledge of the spectrum and Gelfand Theory is used in connection with an idea that relates these operators with multiplication operators on the Hardy space.

September 5

Carl Cowen (IUPUI)

Adjoints of Composition Operators with Rational Symbol (continued)

August 29

Carl Cowen (IUPUI)

Adjoints of Composition Operators with Rational Symbol

The paper of Hammond, Morehouse, and Robbins with the above title will be discussed. Based on carefully chosen examples, the authors divide rational functions into three classes and present results which clarify or extend prior work on adjoints of composition operators on H^2 done by Bourdon, by Martin and Vukotic, and by Gallardo-Gutierrez and the speaker.

SPRING/SUMMER 2007

End of Summer 2007

Spring 2009

Fall 2008

Spring 2008

Fall 2007

Spring 2007

Beginning of Seminar

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July 18

Hermitian Weighted Composition Operators on H^2 (Part II, reprise)

July 11

Carl Cowen (IUPUI)

Hermitian Weighted Composition Operators on H^2 (Part II)

June 27

Carl Cowen (IUPUI)

Hermitian Weighted Composition Operators on H^2

The weighted composition operators that are Hermitian on the Hardy Hilbert space will be identified and their spectral measures will be described. This is joint work of the speaker with Eungil Ko.

June 20

William Higdon (Univ. of Indianapolis)

Ugur Gul's Work on Spectral Theory of Composition Operators on Hardy Spaces

This discussion will be principally based on Ugur Gul's Thesis, February 2007 that may be downloaded above.

End of Spring Semester 2007

Spring 2009

Fall 2008

Spring 2008

Fall 2007

Summer 2007

Beginning of Seminar

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May 9

Carl Cowen (IUPUI)

Adjoints of Composition Operators on the Hardy Hilbert Space: the Rational Function Case, Part IV

April 18

Carl Cowen (IUPUI)

Adjoints of Composition Operators on the Hardy Hilbert Space: the Rational Function Case, Part III

April 11

Carl Cowen (IUPUI)

Adjoints of Composition Operators on the Hardy Hilbert Space: the Rational Function Case, Part II

April 4

Carl Cowen (IUPUI)

Adjoints of Composition Operators on the Hardy Hilbert Space: the Rational Function Case, Part I

We begin with a continuation of last week's discussion of the linear fractional case, finishing the easy proof and giving the hard proof, not originally published but the proof that led to the discovery of the formula in the 1988 paper. These ideas are the foundation of the more recent work on rational function symbols that we will begin in this talk.

March 28

Carl Cowen (IUPUI)

Adjoints of Composition Operators on the Hardy Hilbert Space: the Linear Fractional Case

We begin a discussion of adjoints of composition operators on the space H^2(D), a subject that is not yet closed for general symbols. Formulas have been known for linear fractional symbols and inner function symbols for nearly two decades. More recently, Eva Gallardo-Gutierrez and the speaker have found an interesting formula in the case of rational functions. This week's presentation and those of the next few weeks will be based on the papers "Linear Fractional Composition Operators on H^2" J. Integral Equations and Operator Theory, 11; 151-160, 1988; "The Adjoint of a Composition Operator" preprint, 2005; and "A New Class of Operators and a Description of Adjoints of Composition Operators," J. Functional Analysis, 238; 447-462, 2006.

March 21

Rebecca Wahl (Butler University)

The Weighted Shift Analogy for C*_phi and Spectral Information: phi(z)=z^2/(2-z) (cont'd)

For a certain class of composition operators we show the restriction of C*_phi to an invariant subspace is similar to a weighted shift and that, therefore, the spectrum of C*_phi contains an open disk of eigenvalues. (Based on the paper "Composition operators with multivalent symbol" Acta Scientarium Mathematicarum (Szeged) 66; 755-768, 2000.)

March 7

Rebecca Wahl (Butler University)

The Weighted Shift Analogy for C*_phi and Spectral Information: phi(z)=z^2/(2-z)

For a certain class of composition operators we show the restriction of C*_phi to an invariant subspace is similar to a weighted shift and that, therefore, the spectrum of C*_phi contains an open disk of eigenvalues.

February 28

Carl Cowen (IUPUI)

Introduction to iteration sequences of analytic maps of the disk and to the work of Poggi-Corradini (cont'd)

Will continue discussion of the Poggi-Corradini paper

February 21

Carl Cowen (IUPUI)

Introduction to iteration sequences of analytic maps of the disk and to the work of Poggi-Corradini (cont'd)

Will discuss: inequalities for angular derivatives at fixed points and begin discussion of the Poggi-Corradini paper

February 7

Carl Cowen (IUPUI)

Introduction to iteration sequences of analytic maps of the disk and to the work of Poggi-Corradini (cont'd)

Discussed: conjugation/model for iteration

January 31

Carl Cowen (IUPUI)

Introduction to iteration sequences of analytic maps of the disk and to the work of Poggi-Corradini

Poggi-Corradini's "Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk", Rev. Mat. Iberoamericana 19; 943-970, 2003. (pdf version dvi or ps version: paper #11)
Also useful will be his survey "Iteration of self-maps of the disk: an overview", CUBO 6; 73-80, 2004. (pdf version dvi or ps version : paper #13)
Additional bibliographic items to accompany Poggi-Corradini's paper

Discussed: iteration sequences, interpolation sequences, weighted shifts, adjoint composition operators as weighted shifts, circular symmetry

January 24

Bill Higdon (University of Indianapolis)

The Spectra of Composition Operators from Linear Fractional Maps Acting upon the Dirichlet Space

January 17

Carl Cowen (IUPUI)

Organizational Meeting

Beginning of Seminar

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Spring 2009

Fall 2008

Spring 2008

Fall 2007

Summer 2007

Spring 2007

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