Research

My Research Interests

The study of composition and Toeplitz operators on spaces of analytic functions is an active research area that lies in the intersection of the fields of operator theory, complex analysis, operator algebras, and functional analysis. Both composition and Toeplitz operators are induced by underlying functions, and the properties of the operators are closely tied to the properties of the associated maps. One approach for studying these operators is to investigate the unital C*-algebras generated by collections of the operators. This strategy has been used to analyze Toeplitz operators for over forty years, but its use in the analysis of composition operators only began in the last decade. For this reason, many open questions remain about the structures of unital C*-algebras generated by composition operators.

My research seeks to answer some of these questions in the setting of composition operators acting on the Hardy space of the disk. In my dissertation work, I am investigating the unital C*-algebras generated by collections of composition operators induced by linear-fractional self-maps of the open unit disk ID that fix a given point on the unit circle. I am also studying the C*-algebras generated by these collections together with either the compact operators or the Toeplitz operators induced by continuous functions. The two main goals of my work are:

• to determine the structures and K-theory of the unital C*-algebras generated by these collections of operators and
• to relate the spectrum and essential spectrum of algebraic combinations of these operators to the properties of the underlying functions.

These two aims are interrelated as the structure of the C*-algebra generated by a collection of operators often contains accessible information about the spectra of its elements.

Publications

• Toeplitz-composition C*-algebras induced by linear-fractional non-automorphism self-maps of the disk, submitted.
• Unitary equivalence of composition C*-algebras on the Hardy and weighted Bergman spaces, Complex Anal. Oper. Theory, 10(2016), 1295-1316
• Fixed point composition and Toeplitz-composition C*-algebras, J. Funct. Anal., 265(2013), 743-764.
• A semigroup composition C*-algebra, J. Operator Theory, 67(2012), 581-604.

Conference Research Talks

• C*-algebras generated by composition and Toeplitz operators, Women's Intellectual Network Research Symposium, University of Virginia, Charlottesville, VA, September 2018.
• Linear-fractional composition C*-algebras, Hilbert Function Spaces Conference, Gargnano, Italy, May 2017.
• Composition C*-algebras induced by linear-fractional non-automorphism self-maps of the unit disk, Special Session on Multivariate Operator Theory and Function Theory, AMS Central Sectional Meeting, Indiana University, Bloomington, IN, April 2017. Invited speaker.
• Composition C∗-algebras induced by linear-fractional non-automorphism self-maps of the unit disk, International Workshop on Operator Theory and Applications, Washington University in St. Louis, St. Louis, MO, July 2016.
• Composition C*-algebras, SMPosium, Washington, DC, August 2015
• Unitary equivalence of composition C*-algebras on the Hardy and weighted Bergman spaces, Special Session on Operator Theory on Analytic Function Spaces, AMS Eastern Sectional Meeting, Georgetown University, Washington, DC, March 2015. Invited speaker.
• C*-algebras generated by composition operators on the Hardy and weighted Bergman spaces, Southeastern Analysis Meeting, Clemson University, Clemson, SC, March 2014.
• C*-algebras generated by composition operators on weighted Bergman spaces, Hilbert Functions Spaces Conference, Gargnano, Italy, May 2013.
• C*-algebras generated by composition operators on a weighted Bergman space, Special Session on Operator Theory and Function Spaces, AMS Eastern Sectional Meeting, Rochester Institute of Technology, Rochester, NY, September 2012. Invited Speaker.
• Fixed point composition and Toeplitz-composition C*-algebras,Great Plains Operator Theory Symposium, University of Houston, Houston, TX, May 2012.
• Fixed point composition and Toeplitz-composition C*-algebras, Special Session on Operator Theory and Complex Analysis, AMS Southeastern Sectional Meeting, University of South Florida, Tampa, FL, March 2012. Invited Speaker.
• Composition and Toeplitz-composition C*-algebras related to linear-fractional maps, Special Session on Composition Operators, Canadian Mathematical Society Winter Meeting, Toronto, ON, December 2011. Invited Speaker.
• C*-algebras generated by linear-fractionally-induced composition operators, Southeastern Analysis Meeting, University of Florida, Gainesville, FL, March 2011.
• C*-algebras generated by linear-fractionally-induced composition operators, Special Session on Operator Theory, AMS Southeastern Sectional Meeting, University of Richmond, Richmond, VA, November 2010. Invited speaker.
• Fixed point composition C*-algebras, Joint Mathematics Meetings, San Francisco, CA, January 2010.
• Composition C*-algebras induced by linear-fractional non-automorphisms fixing a boundary point, Virginia Operator Theory and Complex Analysis Meeting<, Virginia Commonwealth University, Richmond, VA, November 2009. Invited speaker.
• A C*-algebra generated by a semigroup of composition operators, Great Plains Operator Theory Symposium, University of Colorado at Boulder, Boulder, CO, June 2009.
• A C*-algebra generated by a semigroup of composition operators, Southeastern Analysis Meeting, University of South Florida, Tampa, FL, March 2009.

Conference Expository Talks

• The Beauty of Analytic Functions, SMPosium 2014, Carleton College, Northfield, MN, July 2014.
• Crocheting a Plane: Using Physical Models to Visualize Hyperbolic Geometry, SMPosium 2013, Carleton College, Northfield, MN, July 2013.
• The Wonders of Infinite-dimensional Space(s): An Introduction to Hilbert Spaces and Bounded Linear Operators , SMPosium 2010, Carleton College, Northfield, MN, July 2010.

Seminar Talks

• University of Virginia, Seminar in Operator Theory and Operator Algebras

  • Visualizing the spectra of composition operators induced by maps in S(2), April 2018.
  • Composition C*-algebras induced by linear-fractional non-automorphism self-maps of the unit disk, November 2016.
  • Unitary equivalence of C*-algebras generated by composition operators on the Hardy and weighted Bergman spaces, April 2014.
  • Unitary equivalence of weighted composition operators on weighted Bergman spaces, November 2012.
  • When is the C*-algebra generated by a C*-algebra and a group of unitary operators isomorphic to a crossed product?, October 2011.
  • Composition operators and crossed products (2 talks), December 2009.
  • An introduction to crossed products of C*-algebras, April 2009.
  • A lower bound on the essential norm of a difference of composition operators, April 2008.
  • On the Toeplitzness of composition operators, October 2006.

• Longwood University, Colloquium, January 2016.

• James Madison University, Institute for Visual Studies Colloquium, February 2013

• Georgia Institute of Technology, Analysis Seminar, October 2012

• Mary Baldwin College, Math Club, March 2012

• Hampden-Sydney College, Colloquium, November 2010

• George Washington University, Analysis Seminar, April 2010

• James Madison University, Colloquium, March 2010

• University of Virginia, Department of Mathematics Graduate Seminar

  • C*-algebras, operators, and almost periodic functions, September 2008.
  • Almost periodic functions, October 2007.
  • An introduction to composition operators, October 2006.

Additional Conferences Attended

• MAA MD-DC-VA Section Meeting
  • University of Mary Washington, Fredericksburg, VA, April 2009, November 2018.
  • Washington and Lee University and Virginia Military Institute, Lexington, VA, April 2018.
  • Christopher Newport University, Newport News, VA, November 2011, November 2017.
  • Frostburg State University, Frostburg, MD, April 2017.
  • Johns Hopkins University, Baltimore, MD, November 2016.
  • Montgomery College, Germantown, MD, April 2016.
  • Roanoke College, Salem, VA, April 2015
  • James Madison University, Harrisonburg, VA, April 2014
  • Virginia Military Institute, Lexington, VA, October 2012
  • Stevenson University, Owings Mills, MD, April 2012
  • Randolph-Macon College, Ashland, VA, April 2011
  • Goucher College, Baltimore, MD, November 2009
• Virginia Operator Theory and Complex Analysis Meeting
  • University of Richmond, Richmond, VA, November 2007, October 2016, October 2018.
  • Washington and Lee University, Lexington, VA, November 2014, November 2015
  • University of Virginia, Charlottesville, VA, October 2013, October 2017.
  • Virginia Commonwealth University, Richmond, VA, September 2010, November 2011
• Project Kaleidoscope (PKAL) STEM Leadership Institute
  • Adamstown, MD, July 2016.
• Shenandoah Undergraduate Mathematics and Statistics Conference
  • James Madison University, Harrisonburg, VA, October 2010, October 2011, September 2012, September 2013, October 2014, Septebmer 2015, October 2017, October 2018.
• Research Experiences for Undergraduate Faculty
  • Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, RI, July 2015
• Uncertainty Principles in Harmonic Analysis: Gap and Type Problems (CBMS Regional Conference in the Mathematical Sciences)
  • Clemson University, Clemson SC, August 2013, Funded Participant
• Southeastern Analysis Meeting
  • Virginia Tech, Blacksburg, VA, March 2013
  • Georgia Institute of Technology, Atlanta, GA, March 2010, Funded Participant
  • University of Richmond, Richmond, VA, March 2007, Funded participant
  • University of Florida, Gainesville, FL, March 2006, Funded participant
• SMPosium
  • Carleton College, Northfield, MN, June/July 2011
• Joint Mathematics Meetings
  • New Orleans, LA, January 2011
  • Washington, DC, January 2009
  • San Diego, CA, January 2008
• Mathfest
  • Pittsburgh, PA, August 2010
• East Coast Operator Algebras Symposium
  • Texas A&M University, College Station, TX, October 2009, Funded Participant
• Career Mentoring Workshop
  • Wheaton College, Norton, MA, July 2009, Funded Participant
• International Workshop on Operator Theory and its Applications
  • College of William and Mary, Williamsburg, VA, July 2008
• Great Plains Operator Theory Symposium
  • University of Cincinnati, Cincinnati, OH, June 2008, Funded Participant
• Classical Analysis Conference
  • University of North Carolina, Chapel Hill, NC, September 2007, Funded Participant

Material from Research that I Conducted as an Undergraduate

• Continuous Nowhere Differentiable Functions, Senior Honors Thesis, University of South Carolina, 2004

This page is maintained by querteks@jmu.edu
Last Updated: December 20, 2018