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Small Toeplitz Operators

Decio, Stefano LU (2018) In Master's Theses in Mathematical Sciences FMAM05 20181
Mathematics (Faculty of Engineering)
Abstract
Toeplitz operators acting on Hilbert spaces of analytic functions are among the most well studied examples of concrete operators. In our work we are interested in a cut-off property of such operators; namely, if the operator is small enough, does it have to be zero? Or more in general, must its symbol be of a particular form? There have been several such results, and in the Hardy space the answer is classical and well known. More recently Daniel Luecking proved such a result in the Bergman space case, with the cutoff being at the finite rank level. We present a new proof of a more general version of that theorem, which unifies several results that followed the publication of Luecking's paper.
Popular Abstract
The first appearance of Toeplitz operators was in the form of Toeplitz matrices, that are matrices which are constant on the diagonals. A Toeplitz operator is an operator whose representation in a basis is an infinite Toeplitz matrix. These operators serve as a concrete model for more general operators in mathematics, and being able to answer questions about Toeplitz operators may shed light on other matters. An example we present in this work is a problem that comes from quantum mechanics, and concerns the motion of a particle confined in a plane under the action of a magnetic field perpendicular to the plane; if a perturbation by an electrostatic potential is introduced, the new states of the particle are analyzed through the spectrum of... (More)
The first appearance of Toeplitz operators was in the form of Toeplitz matrices, that are matrices which are constant on the diagonals. A Toeplitz operator is an operator whose representation in a basis is an infinite Toeplitz matrix. These operators serve as a concrete model for more general operators in mathematics, and being able to answer questions about Toeplitz operators may shed light on other matters. An example we present in this work is a problem that comes from quantum mechanics, and concerns the motion of a particle confined in a plane under the action of a magnetic field perpendicular to the plane; if a perturbation by an electrostatic potential is introduced, the new states of the particle are analyzed through the spectrum of a Toeplitz operator. (Less)
Please use this url to cite or link to this publication:
author
Decio, Stefano LU
supervisor
organization
course
FMAM05 20181
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Operator theory
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFMA-3351-2018
ISSN
1404-6342
other publication id
2018:E33
language
English
id
8946437
date added to LUP
2018-09-21 16:46:26
date last changed
2018-09-21 16:46:26
@misc{8946437,
  abstract     = {Toeplitz operators acting on Hilbert spaces of analytic functions are among the most well studied examples of concrete operators. In our work we are interested in a cut-off property of such operators; namely, if the operator is small enough, does it have to be zero? Or more in general, must its symbol be of a particular form? There have been several such results, and in the Hardy space the answer is classical and well known. More recently Daniel Luecking proved such a result in the Bergman space case, with the cutoff being at the finite rank level. We present a new proof of a more general version of that theorem, which unifies several results that followed the publication of Luecking's paper.},
  author       = {Decio, Stefano},
  issn         = {1404-6342},
  keyword      = {Operator theory},
  language     = {eng},
  note         = {Student Paper},
  series       = {Master's Theses in Mathematical Sciences},
  title        = {Small Toeplitz Operators},
  year         = {2018},
}