Estudiamos estos temas juntos con:

  1. Nikolai Vasilevski (CINVESTAV del IPN),
  2. Crispin Herrera YaƱez (ESCOM del IPN),
  3. Ondrej Hutník (Šafárik University in Košice, Slovakia),
  4. Kevin Michael Esmeral García (Universidad de Sucre, Colombia),
  5. Gerardo Ramos Vázquez (acaba de terminar la maestría en la ESFM del IPN),
  6. Roberto Moisés Barrera Castelán (estudiante de maestría de la ESFM del IPN).

Algunos trabajos recientes

Gerardo Ramos Vázquez, El grupo afín positivo y la transformada de ondícula continua (póster)

Gerardo Ramos Vázquez, La transformada de ondícula continua y y algunas clases de operadores de localización. Tesis de maestría.
Véase también la presentación.


Ejercicios relacionados con operadores de Toeplitz en espacios de Bergman

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Cauchy–Riemann equations (review).
Cambio de variable en una integral de área por medio de una función holomorfa.
Biholomorfismo del disco unitario sobre el semiplano superior.
Möbius transforms of the extended complex plane.
Möbius transforms of the unit disk.
Pseudohyperbolic disk (picture).
Möbius transforms of the upper half-plane (picture).


Weighted Bergman spaces on the unit disk

Varios apuntes de estos temas están escritos por Roberto Moisés Barrera Castelán durante su servicio social.

Teorema de Riesz–Fréchet de representación de funcionales continuos en espacios de Hilbert.
Operador adjunto a un operador acotado que actúa en un espacio de Hilbert.
Espacios de Hilbert con núcleo reproductor.
Orthogonality of the monomials with respect to the weighted measure on the unit disk.
Monomial base in the weighted Bergman space.
Values of analytic functions through area integral.
Reproducing kernel and Bergman projection.
Berezin transform of bounded linear operators.
Analytic functions of two variables vanishing on the conjugate diagonal.
Berezin transform of bounded linear operators is an injective map.
Toeplitz operators generated by bounded symbols.
The correspondence between bounded generating symbols and Toeplitz operators is an injective map.
Berezin transform of bounded functions.


Bounded uniformly continuous functions on a metric space

Recall the definition and some well-known properties of the bounded uniformly continuous complex-valued functions defined on a metric space.

Bounded uniformly continuous functions on a metric space.
Continuous prolongation of a uniformly continuous function defined on a dense subset of a metric space.
Canonical distance on the extended real line.
Continuous functions on the extended real line.


Radial Toeplitz operators on the weighted Bergman space on the unit disk

Multiplication operator on the space of square-summable sequences.
Invertibility and spectrum of the multiplication operator on the space of square-summable sequences.
Criteria of multiplication operator on the space of square-summable sequences.
Criterion of radial operator.
Criterion of Toeplitz radial operator.

Very slowly oscillating functions and their applications to Toeplitz operators

Prove that the spectral functions of the Toeplitz operators generated by vertical symbols are dense in the C*-algebra of the very slowly oscillating functions on the positive half-line.

Logarithmic distance on the positive half-line: ρ(x,y)=|ln(x)−ln(y)|.
Another dilation invariant distance on the positive half-line: ρ(x,y)=|xy|/max(x,y).
Modulus of continuity of functions with respect to the logarithmic distance.
Very slowly oscillating functions on the positive half-line (rough draft).
Some basic properties of the gamma and beta functions (review).
General Leibniz rule (very short review).
Associated Laguerre polynomials: from the Rodrigues' representation to the explicit formula.
Laplace transform of the function (e−t tm)(n).
A special approximate unit.
Another special approximate unit.
The Berezin transform of bounded linear operators is an injective map.
Vertical operators on the Bergman space on the upper half-plane (rough draft).
The correspondence between bounded generating symbols and Toeplitz operators is injective.
Vertical Toeplitz operators on the Bergman space on the upper half-plane (stub).
Vertical Toeplitz operators on the Bergman space on the upper half-plane and very slowly oscillating functions on the positive half-line (rough draft).


References

Bibliographic base (BibTeX)

  1. Nikolai Vasilevski. Commutative Algebras of Toeplitz Operators on the Bergman Space.
    Series: Operator Theory: Advances and Applications, Volume 185, 2008, XXIX, 417 pages, Hardcover.
    ISBN 978-3-7643-8725-9.
    http://www.springer.com/birkhauser/mathematics/book/978-3-7643-8725-9
  2. Kehe Zhu. Operator theory in function spaces. Second edition.
    Series: Mathematical surveys and monographs, Volume 138, 348 pages.
    American Mathematical Society, 2007.
    ISBN 978-0-8218-3965-2.
    http://www.ams.org/bookpages/surv-138
  3. Sergei Grudsky and Nikolai Vasilevski. Bergman-Toeplitz operators: radial component influence.
    Integral Equations and Operator Theory 40 (2001), no. 1, 16–33.
    DOI: 10.1007/BF01202952
    http://www.springerlink.com/content/r36277711r570480
  4. Edmund Landau (1910). Über die Bedeutung einiger neuen Grenzwertsätze der Herren Hardy und Axer.
    Prace Matematyczno-Fizyczne, 21:1, 97–177.
    http://eudml.org/doc/215286
  5. Daniel Suárez. Approximation and symbolic calculus for Toeplitz algebras on the Bergman space.
    Rev. Mat. Iberoamericana 20 (2004), no. 2, 563–610.
    http://projecteuclid.org/euclid.rmi/1087482027
  6. Daniel Suárez. Approximation and the n-Berezin transform of operators on the Bergman space.
    J. reine angew. Math. 581 (2005), 175–192.
    DOI: 10.1515/crll.2005.2005.581.175.
  7. Daniel Suárez. The eigenvalues of limits of radial Toeplitz operators.
    Bull. London Math. Soc. 40 (2008), issue 4, 631–641.
    DOI: 10.1112/blms/bdn042
    http://blms.oxfordjournals.org/content/40/4/631
  8. Nina Zorboska. The Berezin transform and radial operators.
    Proceedings of the American Mathematical Society 131 (2003), no. 3, 793–800.
    DOI: 10.1090/S0002-9939-02-06691-1
    http://www.ams.org/journals/proc/2003-131-03/S0002-9939-02-06691-1/home.html
  9. Boris Korenblum and Kehe Zhu. An application of Tauberian theorems to Toeplitz operators.
    Journal of Operator Theory 33 (1995), issue 2, 353–361.
    http://www.mathjournals.org/jot/1995-033-002/1995-033-002-010.html
  10. Sheldon Axler and Dechao Zheng. The Berezin transform on the Toeplitz algebra.
    Studia Mathematica 127 (1998), 113–136.
    http://www.axler.net/BerezinToeplitz.html
  11. Karel Stroethoff. The Berezin transform and operators on spaces of analytic functions.
    Linear Operators, edited by J. Zemánek, Banach Center Publications 38 (1997), Polish Academy of Sciences, Warsaw, 361–380.
    http://matwbn.icm.edu.pl/ksiazki/bcp/bcp38/bcp38122.pdf
  12. Robert Schmidt. Über divergente Folgen and lineare Mittelbildungen.
    Mathematische Zeitschrift 22 (1925), 89–152.
    DOI: 10.1007/BF01479600
  13. T. Vijayaraghavan. A Tauberian theorem.
    J. London Math. Soc., 1 (1926), 113–120.


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