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Research

RESEARCH INTERESTS:

My research area is in operator theory and complex analysis.  This lies at the cross roads of infinite dimensional linear algebra and analysis. That is, I use real analysis and complex analysis techniques to study infinite matrices. More precisely, I study linear operators on Hilbert space that are "close to normal operators", such as subnormal and hyponormal operators.  For example the cyclic behavior of adjoints of subnormal and hyponormal operators, the dynamics of linear operators and tuples of operators, finding subnormal generators of C*-algebras, and invariant subspaces for the operator of multiplication by z on the Hardy space of the slit disk.  Recently, I have studied convex versions of some of the above topics.  For instance invariant convex sets for linear operators, convex-cyclic linear operators and convex-polynomial approximation.

Publications  
32) Convex Stone-Weierstrass Theorems and Invariant Convex Sets
with Paul McGuire
Rocky Mountain Journal of Mathematics, Volume 49, Number 8 (2019)
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31) Convex-Cyclic Matrices, Convex-Polynomial Interpolation & Invariant Convex Sets
with Paul McGuire
Operators and Matrices Volume 11, Number 2 (2017), 465-492.
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30) On Convex-Cyclic Operators
with Teresa Bermudez and Antonio Bonilla, J.
Math. Anal. Appl. 434 (2016), no. 2, 1166–1181 pdf file
29) Tridiagonal Reproducing Kernels and Subnormality
with Gregory T. Adams and Paul McGuire
J. Operator Theory 70:2 (2013), 477 - 494.
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28) n-Weakly Hypercyclic & n-Weakly Supercyclic Operators
J. Funct. Anal. 263 (2012), no. 8, 2255–2299.
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27) n-Weakly Supercyclic Matrices
Revista R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 105 no. 2, (2011), 433-448.
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26) The State of Subnormal Operators
This article, joint with John B. Conway, appeared in the book A Glimpse at Hilbert Space Operators: Paul Halmos in Memoriam
by Sheldon Axler, Peter Rosenthal, and Donald Sarason
Operator Theory: Advances and Applications, Vol. 207, Springer Basel, 2010
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25) The Hardy Space of a Slit Domain
with Alexandru Aleman and Bill Ross
Frontiers in Mathematics, Birkhauser, 2009
24) Hypercyclic Tuples of Operators and Somewhere Dense Orbits
J. Math. Analysis & Appl. 346 (2008), 82-98.
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23) C*-algebras with Multiple Subnormal Generators
with Paul McGuire
J. Operator Theory 60 (2008), 429-443.
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22) Hypercyclic Pairs of Coanalytic Toeplitz Operators
Integral Eqns & Operator Theory J. 58 (2007), 153-173.
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21) Subnormal and Hyponormal Generators of C*-algebras
with Paul McGuire
J. Functional Analysis 231 (2006), 458-499.
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20) Interpolating Measures for Subnormal Operators
Oberwolfach Reports 2005
1 page summary/abstract of talk given at Oberwolfach, Germany
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19) Some Properties of n-Supercyclic Operators
with Paul Bourdon & Joel H. Shapiro
Studia Mathematica 165 (2004), No. 2, 135-157.
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18) Computing the Fredholm Index of Toeplitz Operators with Continuous Symbols
with Paul McGuire
Proc. Amer. Math. Soc. 133 (2005) 1357-1364.
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17) Linear Chaos?
An elementary introduction to chaos for linear operators
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16) Somewhere Dense Orbits are Everywhere Dense
with Paul Bourdon
Indiana Univ. Math. J. 52 (2003), No. 3, 811-819.
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15) (A Survey of) The Dynamics of Cohyponormal Operators
Contemp. Math. 321 Amer. Math. Soc. (2003), 71-85.
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14) Perturbations of Hypercyclic Vectors
J. Math Analysis & Appl. 273 (2002), No. 1, 67-74
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13) On the Spectral Picture of an Irreducible Subnormal Operator II
with Paul McGuire
Proc. Amer. Math. Soc. 131 (2003), no. 6, 1793--1801.
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12) n-Supercyclic Operators
Studia Mathematica 151 (2002), 141-159.
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11) Hypercyclicity and Supercyclicity for Invertible Bilateral Weighted Shifts
Proc. Amer. Math. Soc. 131 (2003), 479-485.
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10) Countably Hypercyclic Operators
J. Operator Theory 50 (2003), 107-117.
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9) Hypercyclic and Supercyclic Cohyponormal Operators 
with Vivien Miller & Len Miller 
Acta Sci. Math. (Szeged) 68 (2002),  no. 1-2, 303--328.
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8) Pure Subnormal Operators have Cyclic Adjoints
J. Functional Analysis 162 (1999), no. 2, 379 - 399.
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7) Subnormal Operators, Self-Commutators and Pseudocontinuations
Integral Eqns & Operator Theory J. 37 (2000), no. 4, p. 402-422.
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6) Tensor Products of Subnormal Operators
Proc. Amer. Math. Soc. 127 (1999), no. 9, pp. 2685-2695.
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5) Essentially Subnormal Operators
Proc. Amer. Math. Soc. 127 (1999), no. 4, pp. 1171-1181.
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4) Pointwise Multipliers of the Hardy space into the Bergman space
Illinois J. Math. 43 (1999) no. 2, 211-221.
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3) The Berger-Shaw Theorem for Cyclic Subnormal Operators
Indiana Univ. Math. J., 46, No. 3, (1997), p. 741-751.
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2) The Essential Self-Commutator of a Subnormal Operator
with John B. Conway, Proc. Amer. Math. Soc. 125 (1997), p. 243-244.
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1) Major Theorems in Compactness: A Unified approach
with Jerzy Dydak, Amer. Math. Monthly 99 (1992) No. 3, p. 220 - 227.
Original Article