@COMMENT {
Area: Toeplitz operators in the Bergman space.
Written by: Kevin Michael Esmeral García, Crispin Herrera Yañez, Egor Maximenko.
}
ARTICLE { articletemplate,
author = "Surname, First Name",
title = "",
journal = "",
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title = "",
isbn = "",
edition = "Second",
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year = "",
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@ARTICLE { Ahern_Cuckovic_2001_Brown_Halmos_Bergman_Toeplitz,
author = "Ahern, Patrick and {\v{C}}u{\v{c}}kovi{\'c}, {\v{Z}}eljko",
affiliation = "University of Wisconsin, Madison, Wisconsin and University of Toledo, Toledo, Ohio",
title = "A theorem of {B}rown--{H}almos type for {B}ergman space {T}oeplitz operators",
journal = "Journal of Functional Analysis",
issn = "0022-1236(p) 1096-0783(e)",
publisher = "Elsevier",
year = "2001",
volume = "187",
issue = "1",
pages = "200--210",
doi = "10.1006/jfan.2001.3811",
url = "http://www.sciencedirect.com/science/article/pii/S002212360193811X",
mathreviewsid = "1867348 (2002h:47040)",
msc = "Primary: 47B35; Secondary: 30D55",
keywords = "Toeplitz operator, Bergman space, Berezin transform, invariant laplacian",
abstract =
"We study the analogues of the Brown–Halmos theorem for Toeplitz operators on the Bergman space.
We show that for $f$ and $g$ harmonic, $T_f T_g=T_h$ only in the trivial case, provided that $h$ is of class $C^2$ with the invariant laplacian bounded.
Here the trivial cases are $f$ or $g$ holomorphic.
From this we conclude that the zero-product problem for harmonic symbols has only the trivial solution.
Finally, we provide examples that show that the Brown–Halmos theorem fails for general symbols, even for symbols continuous up to the boundary.",
}
@ARTICLE { Axler_Zheng_1998_compact_Berezin,
author = "Axler, Sheldon and Zheng, Dechao",
title = "Compact operators via the {B}erezin transform",
journal = "Indiana Univ. Math. J.",
fulljournal = "Indiana University Mathematics Journal",
issn = "0022-2518",
year = "1998",
volume = "47",
issue = "2",
pages = "387--400",
doi = "10.1512/iumj.1998.47.1407",
url = "http://www.iumj.indiana.edu/IUMJ/fulltext.php?artid=1407&year=1998&volume=47",
mathreviewsid = "1647896 (99i:47045)",
msc = "Primary: 47B35; Secondary: 46E22",
keywords = "?",
abstract = "?",
}
@BOOK { Ahlfors_1979_book_complex_analysis,
author = "Ahlfors, Lars V.",
affiliation = "Harvard University",
title = "Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable",
isbn = "0070006571",
edition = "Second",
publisher = "Third",
year = "1979",
series = "",
volume = "",
msc = "",
keywords = ""
mathreviewsid = "",
abstract = "",
}
@ARTICLE { Axler_Zheng_1998_Berezin_Toeplitz,
author = "Axler, Sheldon and Zheng, Dechao",
title = "The {B}erezin transform on the {T}oeplitz algebra",
journal = "Studia Mathematica",
issn = "0039-3223(p) 1730-6337(e)",
publisher = "Institute of Mathematics Polish Academy of Sciences",
year = "1998",
volume = "127",
issue = "2",
pages = "113--136",
doi = "?",
url = "http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-smv127i2p113bwm",
msc = "47B37",
keywords = "?",
abstract =
"This paper studies the boundary behavior of the Berezin transform
on the $C^\ast$-algebra generated by the analytic Toeplitz operators on the Bergman space.",
}
@ARTICLE { Bargman_1961_analytic_functions_integral_transform_I,
author = "Bargmann, Valentine",
affiliation = "Princeton University",
title = "On a {H}ilbert space of analytic functions and an associated integral transform part {I}",
journal = "Comm. Pure Appl. Math.",
fulljournal = "Communications on Pure and Applied Mathematics",
issn = "1097-0312",
publisher = "Wiley Subscription Services, Inc., A Wiley Company",
year = "1961",
volume = "14",
number = "3",
pages = "187--214",
doi = "10.1002/cpa.3160140303",
url = "http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160140303/abstract",
mathreviewsid = "0157250 (28 #486)",
msc = "?",
keywords = "?",
abstract = "?",
}
@ARTICLE { Bargman_1967_analytic_functions_integral_transform_II,
author = "Bargmann, Valentine",
title = "On a {H}ilbert space of analytic functions and an associated integral transform. {P}art {II}. {A} family of related function spaces application to distribution theory",
journal = "Communications on Pure and Applied Mathematics",
issn = "1097-0312",
publisher = "Wiley Subscription Services, Inc., A Wiley Company",
year = "1967",
volume = "20",
number = "1",
pages = "1--101",
doi = "10.1002/cpa.3160200102",
url = "http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160200102/abstract",
mathreviewsid = "201959 (34 #1836)",
msc = "?",
keywords = "?",
abstract = "?",
}
@ARTICLE{ Bauer_2009_Berezin_Toeplitz_quantization,
author = "Bauer, Wolfram",
affiliation = "Ernst-Moritz-Arndt-Universität Greifswald, Institut für Mathematik und Informatik, Greifswald, Germany",
title = "{B}erezin--{T}oeplitz quantization and composition formulas",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Elsevier",
year = "2009",
volume = "256",
issue = "10",
pages = "3107--3142",
doi = "10.1016/j.jfa.2008.10.002",
url = "http://www.sciencedirect.com/science/article/pii/S0022123608004230",
msc = "?",
keywords = "Berezin--Toeplitz operator, heat equation, Berezin transform, star product",
mathreviewsid = "?",
abstract =
"Extending results in [L.A. Coburn, The measure algebra of the Heisenberg group, J. Funct. Anal. 161 (1999) 509–525;
L.A. Coburn, On the Berezin--Toeplitz calculus, Proc. Amer. Math. Soc. 129 (11) (2001) 3331–3338]
we derive composition formulas for Berezin--Toeplitz operators with i.g. unbounded symbols in the range of certain integral transforms.
The question whether a finite product of Berezin--Toeplitz operators is an operator of this type again
can be answered affirmatively in several cases, but there are also well-known counter examples.
We explain some consequences of such formulas to $C^\ast$-algebras generated by Toeplitz operators.",
}
@BOOK { Bergman_1970_book_kernel_function_and_conformal_mapping,
author = "Bergman, Stefan",
title = "The Kernel Function and Conformal Mapping",
edition = "Second, revised",
publisher = "American Mathematical Society, Providence, R.I.",
year = "1970",
series = "Mathematical Surveys",
volume = "5",
mathreviewsid = "0507701 (58 #22502)",
msc = "Primary 30A31; Secondary 30A30, 32H10",
keywords = "?",
abstract = "?",
}
@ARTICLE { Berezin_1972_symbols,
author = "Berezin, Felix A.",
title = "Covariant and contravariant symbols of operators",
journal = "Mathematics of the USSR Izvestiya",
year = "1972",
volume = "6",
issue = "5",
pages = "1117--1151",
doi = "10.1070/IM1972v006n05ABEH001913",
url = "http://iopscience.iop.org/0025-5726/6/5/A08",
msc = "47A10, 47G05, 47F05, 35P20, 35S15",
keywords = "?",
mathreviewsid = "MR 0350504 (50 #2996)",
abstract = "?",
}
@ARTICLE { Berezin_1975_general_concept_quantization,
author = "Berezin, Felix A.",
affiliation = "Academy of Sciences of the Ukrainian SSR Institute for Theoretical Physics, Kiev, Ukrainian SSR",
title = "General concept of quantization",
yournal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer Berlin / Heidelberg",
year = "1975",
volume = "40",
number = "2",
pages = "153--174",
doi = "10.1007/BF01609397",
url = "http://projecteuclid.org/euclid.cmp/1103860463",
msc = "?",
keywords = "?",
abstract = "?",
}
@ARTICLE { Coburn_2004_Lipschitz_estimate_Berezin_calculus,
author = "Coburn, Luis A.",
affiliation = "Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14260",
title = "A {L}ipschitz estimate for {B}erezin's operator calculus",
journal = "Proc. Amer. Math. Soc.",
fulljournal = "Proceedings of the American Mathematical Society",
issn = "0002-9939(p) 1088-6826(e)",
publisher = "American Mathematical Society",
year = "2005",
volume = "133",
pages = "127--131",
doi = "10.1090/S0002-9939-04-07476-3",
url = "http://www.ams.org/journals/proc/2005-133-01/S0002-9939-04-07476-3",
msc2000 = "Primary 47B32; Secondary 32A36",
keywords = "",
mathreviewsid = "2085161",
abstract =
"F.~A.~Berezin introduced a general ``symbol calculus'' for linear operators on reproducing kernel Hilbert spaces.
For the particular Hilbert space of Gaussian square-integrable entire functions on complex $n$-space,
${\mathbb{C}}^{n}$, we obtain Lipschitz estimates for the Berezin symbols of arbitrary bounded operators.
Additional properties of the Berezin symbol and extensions to more general reproducing kernel Hilbert spaces are discussed.",
}
@ARTICLE { Coburn_2007_sharp_Berezin_Lipschitz_estimates,
author = "Coburn, Luis A.",
affiliation = "Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14260",
title = "Sharp {B}erezin {L}ipschitz estimates",
journal = "Proc. Amer. Math. Soc.",
fulljournal = "Proceedings of the American Mathematical Society",
issn = "0002-9939(p) 1088-6826(e)",
publisher = "American Mathematical Society",
year = "2007",
volume = "135",
pages = "1163--1168",
doi = "10.1090/S0002-9939-06-08569-8",
url = "http://www.ams.org/journals/proc/2007-135-04/S0002-9939-06-08569-8",
msc2000 = "Primary 47B32; Secondary 32A36",
keywords = "?",
mathreviewsid = "2262921",
abstract =
"F.A.~Berezin introduced a general ``symbol calculus'' for linear operators on reproducing kernel Hilbert spaces.
For the Segal-Bargmann space $H^2({\mathbb{C}}^n,d\mu)$ of Gaussian square-integrable entire functions on complex $n$-space, ${\mathbb{C}}^n$,
or for the Bergman spaces $A^2(\Omega)$ of Euclidean volume square-integrable holomorphic functions on bounded domains $\Omega$ in $\mathbb{C}^n$,
we show here that earlier Lipschitz estimates for Berezin symbols of arbitrary bounded operators are sharp.",
}
@BOOK { Conway_1978_book_complex_variable_I,
author = "Conway, John B.",
title = "Functions of One Complex Variable I",
isbn = "0-387-90328-3",
edition = "Second",
publisher = "Springer",
year = "1978",
series = "Graduate texts in mathematics",
volume = "11",
msc = "",
keywords = ""
mathreviewsid = "",
abstract = "",
}
@BOOK { Conway_1985_book_functional_analysis,
author = "Conway, John B.",
affiliation = "Indiana University",
title = "A Course in Functional Analysis",
isbn = "3-540-96042-2",
edition = "Second",
publisher = "Springer-Verlag New York",
year = "1985",
series = "Graduate texts in mathematics",
volume = "96",
msc = "46-01, 45B05",
keywords = ""
mathreviewsid = "",
abstract = "",
}
@BOOK { Deitmar_Echterhoff_2009_book_principles_harmonic_analysis,
author = "Deitmar, Anton and Echterhoff, Siegfried",
affiliation = "Universit{\"{a}}t T{\"{u}}bingen Inst. Mathematik Auf der Morgenstelle 10 72076 T{\"{u}}bingen Germany
and Universit{\"{a}}t M{\"{u}}nster Mathematisches Institut Einsteinstr. 62 48149 M{\"{u}}nster Germany",
title = "Principles of Harmonic Analysis",
publisher = "Springer New York",
series = "Universitext",
volume = "15",
year = "2009",
isbn = "978-0-387-85468-7(p), 978-0-387-85469-4(e)",
doi = "10.1007/978-0-387-85469-4",
msc2000 = "43-01, 42-01, 22Bxx",
pages = "333",
abstract =
"The book is written for graduate students who have read the first book
and like to see the proofs which were not given there and/or want to see the full extent of the theory.
On the other hand it can be read independently from the first one,
only a modest knowledge on Fourier series/tranform is required to understand the examples.
This book fills a major gap in the textbook literature,
as a full proof of Pontryagin Duality and Plancherel Theorem is hard to come by.
It is usually given in books that focus on C*-algebras and thus carry too much technical overload for the reader
who only wants these basic results of Harmonic Analysis.
Other proofs use the structure theory which carries the reader away in a different direction.
Here the authors consider the Banach-algebra approach more elegant and enlighting.
They provide a streamlined approach that reaches the main results directly,
and they also give the generalizations to the non-Abelian case.
Another main pillar of Harmonic analysis is the Poisson Summation Formula.
We give its generalization to LCA-groups.
The Selberg Trace Formula is considered the generalization of the Poisson Formula to non-abelian groups.
The authors give the first textbook approach to this deep and useful formula in full generality.
The last two chapters are devoted to examples of applications of the Selberg Trace Formula.",
}
@ARTICLE{ Englis_1990_density_Toeplitz_Bergman,
author = "Engli{\v{s}}, Miroslav",
title = "Some density theorems for {T}oeplitz operators on {B}ergman spaces",
journal = "Czechoslovak Mathematical Journal",
issn = "0011-4642",
publisher = "Institute of Mathematics, Academy of Sciences of the Czech Republic Matematick{\'{y}} {\'{u}}stav AV {\v{C}}R",
year = "1990",
volume = "40",
issue = "3",
pages = "491--502",
doi = "?",
url = "http://dml.cz/dmlcz/102402",
mathreviewsid = "1065029",
zentralblattmath = "0736.47018",
msc = "47B35, 47D99",
keywords = "?",
abstract =
"The set of all Toeplitz operators on the Bergman space of the unit disc
is shown to be dense in the set of all bounded linear operators,
in the weak operator topology.
Some results concerning norm density are also given.",
}
@ARTICLE{ Englis_1996_Berezin_quantization_reproducing_kernels,
author = "Engli{\v{s}}, Miroslav",
title = "{B}erezin quantization and reproducing kernels on complex domains",
journal = "Transactions of the American Mathematical Society",
issn = "?",
publisher = "?",
year = "1996",
volume = "348",
number = "2",
pages = "?",
doi = "?",
url = "?",
msc = "?",
keywords = "quantization, Berezin transform, weighted Bergman spaces,
covariant symbols of operators, reproducing (Bergman) kernels, asymptotic behaviour,
pseudoconvex domains, complex ellipsoids, K{\"{a}}ler-Einstein metric",
abstract = "?",
}
@COMMENT{
TECHREPORT{mecanica,
AUTHOR = {Rodrigo Ferrer P., Herbert Massmann L., Jaime Roessler B., José Rogan C.},
TITLE = {MEC\'ANICA CU\'ANTICA I},
INSTITUTION = {Departamento de Física, Facultad de Ciencias, Universidad de Chile.},
address = {secretaria@física.ciencias.uchile.cl},
keywords = {Mecánica Cuántica},
}
}
@BOOK { Garnett_1981_book_bounded_analytic_functions,
author = "Garnett, John B.",
title = "Bounded Analytic Functions",
edition = "Revised 1st",
year = "2007",
series = "Graduate Texts in Mathematics",
volume = "236",
publisher = "Springer",
doi = "10.1007/0-387-49763-3",
url = "http://www.springer.com/mathematics/analysis/book/978-0-387-33621-3",
msc = "?",
keywords = "?",
abstract = "?",
}
@ARTICLE { Gautrin_1988_Toeplitz_Bargmann,
author = "Gautrin, Henri-Fran{\c{c}}ois",
affiliation = "Département de mathématiques et de statistique Université de Montréal Montréal Québec Canada",
title = "{T}oeplitz operators in {B}argmann spaces",
journal = "Integr. Equ. Oper. Theory",
fulljournal = "Integral Equations and Operator Theory",
issn = "0378-620X(p) 1420-8989(e)",
publisher = "Birkh{\"{a}}user Basel",
year = "1988",
volume = "11",
number = "2",
pages = "173--185",
doi = "10.1007/BF01272117",
url = "http://www.springerlink.com/content/j9mt140h66262l0u",
msc = "47B35",
keywords = "Bargmann space, bounded Toeplitz operators, Toeplitz operator",
mathreviewsid = "",
zentralblattmath = "0644.47028",
abstract = "In this paper we study the notion of Toeplitz operators in a Bargmann space;
more precisely it is shown that every bounded operator is the uniform limit of Toeplitz operators.
We generalize the definition of a Toeplitz operator and show that a large class of operators,
which includes the bounded operators, are ``generalized Toeplitz operators''.",
}
@BOOK { Grafakos_2008_book_classical_Fourier_analysis,
author = "Grafakos, Loukas",
title = "Classical {F}ourier Analysis",
edition = "Second",
publisher = "Springer",
year = "2008",
series = "Graduate Text in Mathematics",
volume = "249",
pages = " ",
isbn = "978-0-387-09431-1",
doi = "10.1007/978-0-387-09432-8",
url = "http://www.springer.com/mathematics/analysis/book/978-0-387-09431-1, http://www.springerlink.com/content/978-0-387-09431-1",
msc2000 = "42-xx 42-02",
keywords = "?",
abstract = "?",
}
@ARTICLE { Grudsky_Vasilevski_2001_Bergman_Toeplitz_radial,
author = "Grudsky, Sergei M. and Vasilevski, Nikolai L.",
title = "{B}ergman-{T}oeplitz operators: radial component influence",
journal = "Integr. Equ. Oper. Theory",
fulljournal = "Integral Equations and Operator Theory",
issn = "0378-620X(p) 1420-8989(e)",
publisher = "Birkh{\"{a}}user Basel",
year = "2001",
volume = "40",
number = "1",
pages = "16--33",
doi = "10.1007/BF01202952",
url = "http://www.springerlink.com/content/r36277711r570480",
msc = "Primary: 47B35; Secondary: 47D25, 46E15, 47A10, 47B07, 47L80",
keywords = "",
mathreviewsid = "1829512 (2002g:47053)",
abstract =
"We analyze the influence of the radial component of a symbol to spectral,
compactness, and Fredholm properties of Toeplitz operators, acting on the Bergman space.
We show that there exist compact Toeplitz operators whose (radial) symbols are unbounded near the unit circle $\partial\mathbf{D}$.
Studying this question we give several sufficient, and necessary conditions, as well as the corresponding examples.
The essential spectra of Toeplitz operators with pure radial symbols have sufficiently rich structure, and even can be massive.
The $C*$-algebras generated by Toeplitz operators with radial symbols are commutative, but the semicommutators
$[T_a, T_b)=T_a\cdot T_b–T_{a\cdot b}$ are not compact in general.
Moreover for bounded operators $T_a$ and $T_b$ the operator $T_{a\cdot b}$ may not be bounded at all.",
}
@ARTICLE { Grudsky_Karapetyants_Vasilevski_2003_unit_ball_radial,
author = "Grudsky, Sergei M. and Karapetyants, Alexei N. and Vasilevski, Nikolai L.",
title = "{T}oeplitz operators on the unit ball in $\mathbb{C}^n$ with radial symbols",
journal = "J.~Operator Theory",
fulljournal = "Journal of Operator Theory",
issn = "1841-7744(e)",
publisher = "The Theta Foundation, Bucharest",
year = "2003",
volume = "49",
issue = "",
pages = "325--346",
doi = "",
url = "",
msc2000 = "32A36, 47B35",
keywords = "weighted Bergman space, Toeplitz operators",
mathreviewsid = "",
abstract =
"The paper is devoted to the study of Toeplitz operators with radial symbols
on the weighted Bergman spaces on the unit ball in ${\mathbb{C}}^n$.
Admitting ``badly'' behaved unbounded symbols we get new qualitative features.
In particular, contrary to known results, a Toeplitz operator with the same (unbounded) symbol
now can be bounded in one weighted Bergman space and unbounded in another,
compact in one weighted Bergman space and bounded but not compact in another,
compact in one weighted Bergman space and unbounded in another.
In our case of radial symbols, the Wick (or covariant) symbol of a Toeplitz operator
gives complete information about the operator, providing its spectral decomposition.",
}
@ARTICLE { Grudsky_Karapetyants_Vasilevski_2004_dynamics_radial,
author = "Grudsky, Sergei M. and Karapetyants, Alexei N. and Vasilevski, Nikolai L.",
title = "Dynamics of properties of {T}oeplitz operators with radial symbols",
journal = "Integr. Equ. Oper. Theory",
fulljournal = "Integral Equations and Operator Theory",
issn = "0378-620X(p) 1420-8989(e)",
publisher = "Birkh{\"{a}}user Verlag Basel/Switzerland",
year = "2004",
volume = "50",
number = "2",
pages = "217--253",
doi = "10.1007/s00020-003-1295-z",
url = "http://www.springerlink.com/content/qwb3lfh68c7p7f4b/",
msc2000 = "Primary 47B35; Secondary 47A25, 47B10, 46E22",
keywords = "Toeplitz operator, weighted Bergman spaces, boundedness, compactness, spectra",
mathreviewsid = "",
abstract =
"In the case of radial symbols we study the behavior of different properties
(boundedness, compactness, spectral properties, etc.) of Toeplitz operators $T_a^{(\lambda)}$
acting on weighted Bergman spaces ${\mathcal{A}}^2({\mathbb{D}})$ over the unit disk ${\mathbb{D}}$, in dependence on $\lambda$,
and compare their limit behavior under $\lambda\to+\infty$ with corresponding properties of the initial symbol $a$.",
}
@ARTICLE { Grudsky_Karapetyants_Vasilevski_2004_dynamics_parabolic,
author = "Grudsky, Sergei M. and Karapetyants, Alexei N. and Vasilevski, Nikolai L.",
title = "Dynamics of properties of {T}oeplitz operators on the upper half-plane: parabolic case",
journal = "J.~Operator Theory",
fulljournal = "Journal of Operator Theory",
issn = "1841-7744(e)",
publisher = "The Theta Foundation, Bucharest",
year = "2004",
volume = "52",
issue = "1",
pages = "1--31",
doi = "",
url = "http://www.mathjournals.org/jot/2004-052-001/2004-052-001-011.html",
msc = "",
keywords = "",
mathreviewsid = "",
abstract =
"We consider Toeplitz operators $T_a^{(\lambda)}$
acting on the weighted Bergman spaces ${\mathcal{A}}_\lambda^2(\Pi)$, $\lambda \in [0,\infty)$,
over the upper half-plane $\Pi$, whose symbols depend on $y=\mathoperator{im}z$.
Motivated by the Berezin quantization procedure we study
the dependence of the properties of such operators on the weight $\lambda$
and, in particular, under the limit procedure $\lambda\rightarrow\infty$.",
}
@ARTICLE { Grudsky_Karapetyants_Vasilevski_2004_dynamics_hyperbolic,
author = "Grudsky, Sergei M. and Karapetyants, Alexei N. and Vasilevski, Nikolai L.",
title = "Dynamics of properties of {T}oeplitz operators on the upper half-plane: hyperbolic case",
journal = "Bol. Soc. Mat. Mexicana",
issn = "",
publisher = "",
year = "2004",
volume = "10",
number = "2",
pages = "",
doi = "",
url = "",
msc = "",
keywords = "Toeplitz operators, weighted Bergman spaces, boundedness, spectrum",
mathreviewsid = "",
abstract =
"We consider Toeplitz operators $T_a^{(\lambda)}$
acting on the weighted Bergman spaces ${\mathcal{A}}_\lambda^2(\Pi)$, $\lambda\in[0,\infty)$,
over the upper half-plane $\Pi$, whose symbols depend on $\theta=\arg z$.
Motivated by the Berezin quantization procedure
we study the dependence of the properties of such operators on the parameter of the weight $\lambda$¸
and, in particular, under the limit $\lambda\to\infty$.",
}
@ARTICLE { Guillemin_1984_Toeplitz_n_dimensions,
author = "Guillemin, Victor",
title = "{T}oeplitz operators in n-dimensions",
journal = "Integr. Equ. Oper. Theory",
fulljournal = "Integral Equations and Operator Theory",
issn = "0378-620X(p) 1420-8989(e)",
publisher = "Birkh{\"{a}}user Basel",
year = "1984",
volume = "7",
number = "2",
pages = "145--205",
mathreviewsid = "750217 (86i:58130)",
doi = "10.1007/BF01200373",
url = "http://www.springerlink.com/content/w617vh25h5754rg3",
msc = "?",
keywords = "?",
abstract =
"The interplay between the theory of Toeplitz operators on the circle
and the theory of pseudodifferential operators on the line (i. e. Wiener-Hopf operators)
is by now well-known and well-understood.
In this article we show that there is a parallel situation in higher dimensions.
To begin with, by using pseudodifferential multipliers, one can simplify the composition rules for Toeplitz operators, (\S~3),
and describe precisely how Toeplitz operators of ``Bergmann type''
are related to Toeplitz operators of ``Szeg{\"{o}} type'' (\S~9).
Furthermore, it turns out that the ring of pseudodifferential operators on a compact manifold, M,
is isomorphic with the ring of Toeplitz operators on an appropriate Grauert tube about M (\S~\S~4--6),
and the ring of Weyl operators on $\mathbb{R}^n$ is isomorphic
with the ring of Toeplitz operators on the complex ball in $\mathbb{C}^n$ (\S~\S~7–-10)."
}
@BOOK { Halmos_1967_book_Hilbert_space_problem_book,
author = "Halmos, Paul R.",
title = "A {H}ilbert Space Problem Book",
isbn = "",
edition = "",
publisher = "D. Van Nostrand Company, Inc., Princeton, New Jersey",
year = "1967",
series = "",
volume = "",
msc = "",
keywords = ""
mathreviewsid = "",
abstract = "",
}
@BOOK { Hille_1982_book_analytic_function_theory_I,
author = "Hille, Einar",
title = "Analytic Function Theory, I",
isbn = "0-8284-0269-8",
edition = "Second",
publisher = "Chelsea Publishing Company, New York",
year = "1982",
series = "",
volume = "",
msc = "",
keywords = ""
mathreviewsid = "",
abstract = "",
}
@ARTICLE { Hormander_1960_estimates,
author = "H{\"{o}}rmander, Lars",
affiliation = "Stockholm",
title = "Estimates for translation invariant operators in $L^p$ spaces",
journal = "Acta Mathematica",
issn = "0001-5962",
publisher = "Springer",
year = "1960",
volume = "104",
number = "1",
pages = "93--140",
doi = "10.1007/BF02547187",
url = "http://www.springerlink.com/content/6473j22054706403",
msc = "?",
keywords = "?",
mathreviewsid = "?",
abstract = "?",
}
@ARTICLE { Jiang_Peng_1992_Toeplitz_and_Hankel_upper_half_plane,
author = "Jiang, Qintang and Peng, Lizhong",
title = "{T}oeplitz and {H}ankel type operators on the upper half plane",
journal = "Integr. Equ. Oper. Theory",
fulljournal = "Integral Equations and Operator Theory",
issn = "0378-620X(p) 1420-8989(e)",
publisher = "Birkh{\"{a}}user Basel",
year = "1992",
volume = "15",
pages = "744--767",
doi = "0378-620X/92/050744-24",
url = "?",
msc = "?",
keywords = "?",
mathreviewsid = "?",
abstract = "?",
}
@ARTICLE { Korenblum_Zhu_1995_Tauberian_Toeplitz_radial,
author = "Korenblum, Boris and Zhu, Kehe",
title = "An application of {T}auberian theorems to {T}oeplitz operators",
journal = "Journal of Operator Theory",
year = "1995",
volume = "33",
issue = "2",
pages = "353–-361",
doi = "?",
url = "http://www.mathjournals.org/jot/1995-033-002/1995-033-002-010.html",
msc = "Primary 47B35; Secondary 47B07, 40E05, 44A15, 44A60",
keywords = "Tauberian theorems, compact Toeplitz operators, Bergman space",
mathreviewsid = "?",
abstract = "?",
}
@BOOK { Krantz_2001_book_several_complex_variables,
author = "Krantz, Steven G.",
title = "Function Theory of Several Complex Variables",
edition = "Second",
publisher = "American Mathematical Society",
year = "2001",
series = "AMS Chelsea Publishing Series",
isbn13 = "9780821827246",
isbn = "0821827243",
mathreviewsid = "635928 (84c:32001)",
abstract =
"The theory of several complex variables can be studied from several different perspectives.
In this book, Steven Krantz approaches the subject from the point of view of a classical analyst, emphasizing its function-theoretic aspects.
He has taken particular care to write the book with the student in mind, with uniformly extensive and helpful explanations,
numerous examples, and plentiful exercises of varying difficulty.
In the spirit of a student-oriented text, Krantz begins with an introduction to the subject,
including an insightful comparison of analysis of several complex variables with the more familiar theory of one complex variable.
The main topics in the book include integral formulas, convexity and pseudoconvexity, methods from harmonic analysis,
and several aspects of the $\overline{\partial}$ problem.
Some further topics are zero sets of holomorphic functions, estimates, partial differential equations, approximation theory,
the boundary behavior of holomorphic functions, inner functions, invariant metrics, and holomorphic mappings.
While due attention is paid to algebraic aspects of several complex variables (sheaves, Cousin problems, etc.),
the student with a background in real and complex variable theory, harmonic analysis, and differential equations will be most comfortable with this treatment.
This book is suitable for a first graduate course in several complex variables.",
}
@ARTICLE { Landau_1910_Bedeutung_Grentzwertsaetze,
author = "Landau, Edmund",
affiliation = "",
title = "{\"{U}}ber die {B}edeutung einiger neuen {G}rentzwerts{\"{a}}tze der {H}erren {H}ardy und {A}xer",
journal = "Prace Mat. Fiz.",
issn = "",
publisher = "",
year = "1910",
volume = "21",
issue = "",
pages = "97--177",
doi = "?",
url = "",
msc = "",
keywords = "?",
mathreviewsid = "",
abstract = "?",
}
@ARTICLE { Luecking_1987_trace_ideal_criteria_Toeplitz,
author = "Luecking, Daniel H.",
affiliation = "Department of Mathematics, University of Arkansas",
title = "Trace ideal criteria for {T}oeplitz operators",
journal = "Journal of Functional Analysis",
issn = "0022-1236(p) 1096-0783(e)",
publisher = "Elsevier",
year = "1987",
volume = "73",
issue = "2",
pages = "345--368",
doi = "10.1016/0022-1236(87)90072-3",
url = "http://www.sciencedirect.com/science/article/pii/0022123687900723",
msc = "?",
keywords = "?",
abstract = "?",
}
@ARTICLE { Luecking_2008_finite_rank_Toeplitz_Bergman,
author = "Luecking, Daniel H.",
affiliation = "Department of Mathematical Sciences, University of Arkansas",
title = "Finite rank {T}oeplitz operators on the {B}ergman space",
journal = "Proc. Amer. Math. Soc.",
issn = "0002-9939(p) 1088-6826(e)",
publisher = "American Mathematical Society",
year = "2008",
volume = "136",
pages = "1717--1723",
doi = "10.1090/S0002-9939-07-09119-8"
msc2000 = "Primary 46E20",
keywords = "Bergman space, Toeplitz operator",
abstract = "?",
}
@ARTICLE { Lusky_1998_approximation_Toeplitz,
author = "Lusky, Wolfgang",
title = "On approximation by {T}oeplitz operators",
journal = "Acta Universitatis Carolinae. Mathematica et Physica",
issn = "?",
publisher = "?",
year = "1998",
volume = "39",
issue = "1--2",
pages = "137--146",
doi = "",
url = "",
msc = "",
keywords = "",
mathreviewsid = "",
abstract = "We show that the set of compact Toeplitz operators
is dense in the space of all compact operators for many generalized Bergman-Hardy spaces.
Moreover the set of $p$-Schatten class Toeplitz operators
is dense in the $p$-Schatten class with respect to the $p$-Schatten class norm for $p$ > 1.",
}
@ARTICLE { McDonald_Sundberg_1979_Toeplitz_disc,
author = "McDonald, G and Sundberg, C",
title = "{T}oeplitz operators on the disc",
journal = "Indiana Univ. Math. J.",
fulljournal = "Indiana University Mathematics Journal",
issn = "?",
publisher = "?",
year = "1979",
volume = "28",
number = "4",
pages = "595--611",
doi = "10.1512/iumj.1979.28.28042",
url = "http://www.iumj.indiana.edu/docs/28042/28042.asp",
msc = "?",
keywords = "?",
}
@ARTICLE { Miao_Zheng_2004_compact_Bergman,
author = "Miao, Jie and Zheng, Dechao",
title = "Compact operators on {B}ergman spaces",
journal = "Integral Equations and Operator Theory",
issn = "",
publisher = "",
year = "2004",
volume = "48",
issue = "1",
pages = "61--79",
doi = "10.1007/s00020-002-1176-x",
url = "http://www.springerlink.com/content/udfgvctdcatxphuf",
msc2000 = "Primary: 47B35; Secondary: 46E15",
keywords = "Toeplitz operator, compact operator, Beresin transform",
mathreviewsid = "",
abstract =
"We prove that a bounded operator $S$ on $L_a^p$ for $p>1$ is compact
if and only if the Berezin transform of $S$ vanishes on the boundary of the unit disk
if $S$ satisfies some integrable conditions.
Some estimates about the norm and essential norm of Toeplitz operators with symbols in BT are obtained.",
}
@ARTICLE { Nam_Zheng_Zhong_2006_mBerezincompact,
author = "Nam, Kyesook and Zheng, Dechao and Zhong, Changyong",
title = "$m$-{B}erezin transform and compact operators",
journal = "Rev. Mat. Iberoamericana",
fulljournal = "Revista Matem{\'{a}}tica Iberoamericana",
publisher = "European Mathematical Society Publishing House",
year = "2006",
volume = "22",
issue = "3",
pages = "867--892",
doi = "10.4171/RMI/477",
url = "http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=22&iss=3&rank=5",
msc = "?",
keywords = "$m$-Berezin transforms, Toeplitz operators",
abstract =
"m-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball.
The norm of the m-Berezin transform as a linear operator from the space of bounded operators to $L^\infty$ is found.
We show that the m-Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball.
Using the m-Berezin transforms we show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the unit ball.",
}
@BOOK { Natanson_1961_book_theory_functions,
author = "Natanson, Isidor Pavlovich",
title = "Theory of Functions of a Real Variable",
isbn10 = "0804447039",
isbn13 = "978-0804447034",
publisher = "Frederick Ungar Publishing Co., New York",
year = "1961",
volume = "I",
isbn = "?",
url = "?",
msc = "?",
keywords = "?",
}
@BOOK { Okikio_1971_book_Lp_spaces,
author = "Okikiolu, George O.",
title = "Aspects of the Theory of Bounded Integral Operators in {L}p-Spaces",
isbn = "978-0-1252-5150-1",
publisher = "Academic Press",
year= "1971",
url = "http://www.abebooks.com/products/isbn/9780125251501",
mathreviewsid = "?",
msc = "?",
keywords = "?",
}
@ARTICLE { Peetre_1990_Berezin_Haplitz,
author = "Peetre, Jaak",
title = "The {B}erezin transform and {H}a-plitz operators",
journal = "J.~Operator Theory",
issn = "?",
publisher = "?",
year = "1990",
volume = "24",
pages = "165--186",
doi = "?",
url = "http://www.theta.ro/jot/archive/1990-024-001/1990-024-001-013.pdf",
msc = "?",
keywords = "?",
mathreviewsid = "?",
abstract = "?",
}
@BOOK { Rudin_1980_book_unit_ball,
author = "Rudin, Walter",
title = "Function Theory in the Unit Ball in $\mathbb{C}^n$",
year = "1980",
publisher = "Springer-Verlag, Berlin and New York",
isbn = "?",
series= "Grundlehren der mathematischen Wissenschaften",
volume = "241",
}
@BOOK { Rudin_1987_book_real_and_complex_analysis,
author = "Rudin, Walter",
title = "Real And Complex Analysis",
edition = "third",
year = "1987",
publisher = "McGraw-Hill, New York",
isbn13 = "978-0-07-054234-1",
}
@BOOK { Rudin_1991_book_functional_analysis,
author = "Rudin, Walter",
title = "Functional Analysis",
isbn13 = "978-0070542365",
isbn = "",
edition = "",
publisher = "McGraw-Hill",
year = "1991",
series = "",
volume = "",
msc = "",
keywords = ""
mathreviewsid = "",
abstract = "",
}
@BOOK { Rudin_2008_book_unit_ball,
author = "Rudin, Walter",
title = "Function Theory in the Unit Ball in $\mathbb{C}^n$",
edition = "Reprint of the 1980 Edition",
year = "2008",
publisher = "Springer-Verlag, Berlin and New York",
isbn = "978-3-540-68272-1(p), 978-3-540-68276-9(e)",
series= "Classics in Mathematics",
volume = "",
doi = "10.1007/978-3-540-68276-9",
url = "http://www.springer.com/mathematics/analysis/book/978-3-540-68272-1",
msc2000 = "32-02, 31Bxx, 31Cxx, 32Axx, 32Fxx, 32Hxx",
}
@ARTICLE { Schmidt_1925_divergente_Folgen,
author = "Schmidt, Robert",
affiliation = "K{\"{o}}nigsberg",
title = "{\"{U}}ber divergente {F}olgen and lineare {M}ittelbildungen",
journal = "Math. Z.",
fulljournal = "Mathematische Zeitschrift",
issn = "0025-5874",
publisher = "Springer Berlin / Heidelberg",
year = "1925",
volume = "22",
issue = "1",
pages = "89--152",
doi = "10.1007/BF01479600",
url = "http://dx.doi.org/10.1007/BF01479600",
msc = "",
keywords = "?",
mathreviewsid = "",
abstract = "?",
}
@ARTICLE { Stanojevic_Stanojevic_2002_Tauberian_retrieval_theory,
author = "Stanojevi{\'{c}}, {\v{C}}aslav V. and Stanojevi{\'{c}}, Vera B.",
title = "Tauberian retrieval theory",
journal = "Publications de l'Institut Mathematique",
issn = "",
publisher = "",
year = "2002",
volume = "71",
issue = "85",
pages = "105--111",
doi = "10.2298/PIM0271105S",
url = "",
msc = "",
keywords = "Karamata's Hauptsatz, controls of oscillatory behavior, retrieval of manageable divergent processes",
mathreviewsid = "",
abstract =
"Karamata's Hauptsatz [11] and its corollary is the main tool
for convergence recovery from Abel's necessary conditions and the control of oscillatory behavior of limiting processes.
By modifying the basic discovery from [11], relaxing Abel's necessary conditions
and lightening the control of oscillatory behavior an extended Tauberian theory is outlined.
This theory goes beyond convergence recovery.
It retrieves various kinds of moderate divergence.",
}
@ARTICLE { Stroethoff_1997_Berezin_transform,
author = "Stroethoff, Karel",
title = "The {B}erezin transform and operators on spaces of analytic functions",
journal = "Linear Operators, Banach Center Publications",
issn = "0137-6934",
publisher = "Institute of Mathematics Polish Academy of Sciences, Warsaw",
year = "1997",
volume = "38",
issue = "1",
pages = "361--380",
doi = "?",
url = "http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-bcpv38i1p361bwm",
msc = "Primary 47B35; Secondary 47B38",
keywords = "?",
mathreviewsid = "1457018 (98g:47025)",
abstract = "?",
}
@ARTICLE { Suarez_2004_Toeplitz_Bergman_commutator_ideal,
author = "Su{\'{a}}rez, Daniel",
title = "The {T}oeplitz algebra on the {B}ergman space coincides with its commutator ideal",
journal = "J.~Operator Theory",
issn = "???",
publisher = "???",
year = "2004",
volume = "51",
number = "1",
pages = "105--114",
doi = "?",
url = "?",
msc = "?",
keywords = "commutator ideal, Toeplitz operators, Bergman space",
abstract = "?",
}
@ARTICLE { Suarez_2004_approximation_Toeplitz_Bergman,
author = "Su{\'{a}}rez, Daniel",
title = "Approximation and symbolic calculus for {T}oeplitz algebras on the {B}ergman space",
journal = "Rev. Mat. Iberoamericana",
fulljournal = "Revista Matem{\'{a}}tica Iberoamericana",
publisher = "European Mathematical Society Publishing House",
year = "2004",
volume = "20",
number = "2",
pages = "563--610",
doi = "?",
url = "http://projecteuclid.org/euclid.rmi/1087482027",
mathreviewsid = "2073132",
zentralblattmath = "02110199",
msc = "Primary 32A36; Secondary 47B35",
keywords = "Bergman space, Toeplitz operator, commutator ideal and abelianization",
abstract =
"If $f\in L^\infty(\mathbb{D})$ let $T_f$ be the Toeplitz operator on the Bergman space $L^2_a$ of the unit disk $\mathbb{D}$.
For a $C^\ast$-algebra $A\subset L^\infty(\mathbb{D})$ let $\mathfrak{T}(A)$ denote the closed operator algebra generated by $\{ T_f : f\in A \}$.
We characterize its commutator ideal $\comm(A)$ and the quotient $\mathfrak{T}(A)/ \mathfrak{C}(A)$ for a wide class of algebras $A$.
Also, for $n\geq 0$ integer, we define the $n$-Berezin transform $B_nS$ of a bounded operator $S$,
and prove that if $f\in L^\infty(\mathbb{D})$ and $f_n = B_n T_f$ then $T_{f_n} \rightarrow T_f$.",
}
@ARTICLE { Suarez_2005_n_Berezin_transform,
author = "Su{\'{a}}rez, Daniel",
affiliation = "Departament de Matem{\`{a}}tiques, Universitat Aut{\`{o}}noma de Barcelona, 08193, Bellaterra, Barcelona, Spain",
title = "Approximation and the $n$-{B}erezin transform of operators on the {B}ergman space",
journal = "J. reine angew. Math.",
fulljournal = "Journal f{\"{u}}r die reine und angewandte Mathematik",
issn = "0075-4102(p), 1435-5345(e)",
publisher = "Walter de Gruyter, Berlin -- New York",
year = "2005",
volume = "581",
pages = "175--192",
doi = "10.1515/crll.2005.2005.581.175",
url = "http://www.degruyter.com/view/j/crll.2005.2005.issue-581/crll.2005.2005.581.175/crll.2005.2005.581.175.xml",
mathreviewsid = "?",
msc = "?",
keywords = "?",
abstract =
"To any bounded operator $S$ on the Bergman space $L_a^2$
we associate a sequence of linear transforms $B_n(S)\in L^\infty(D)$, where $n\ge0$,
and prove that the Toeplitz operators tend to $S$ for some especial classes of operators $S$.
In particular, this holds for every radial operator in the Toeplitz algebra.
Finally, we show that the inclusion of the Toeplitz algebra into the essential commutant of the Bergman shift is proper.",
}
@ARTICLE { Suarez_2008_eigenvalues_radial_Toeplitz,
author = "Su{\'{a}}rez, Daniel",
affiliation = "Departament de Matemàtiques, Universitat Autònoma de Barcelona",
title = "The eigenvalues of limits of radial {T}oeplitz operators",
journal = "Bull. London Math. Soc.",
fulljournal = "Bulletin of the London Mathematical Society",
issn = "0024-6093(p), 1469-2120(e)",
publisher = "London Mathematical Society",
year = "2008",
volume = "40",
number = "4",
pages = "631--641",
doi = "10.1112/blms/bdn042",
url = "http://blms.oxfordjournals.org/content/40/4/631",
msc = "Primary: 32A36; Secondary: 47L80",
keywords = "?",
mathreviewsid = "?",
abstract =
"Let $A^2$ be the Bergman space on the unit disk.
A bounded operator $S$ on $A^2$ is called radial if $S z^n = λ_n z^n$ for all $n\ge0$,
where $λ_n$ is a bounded sequence of complex numbers.
We characterize the eigenvalues of radial operators that belong to the Toeplitz algebra.",
}
@ARTICLE { Vasilevski_1999_Bergman_Toeplitz_commutative,
author = "Vasilevski, Nikolai L.",
title = "On {B}ergman-{T}oeplitz operators with commutative symbol algebras",
journal = "Integr. Equ. Oper. Theory",
fulljournal = "Integral Equations and Operator Theory",
issn = "0378-620X",
publisher = "Birkh{\"{a}}user-Verlag",
language = "English",
year = "1999",
volume = "34",
issue = "1",
pages = "107--126",
doi = "10.1007/BF01332495",
url = "http://www.springerlink.com/content/uq6482332275501r, http://rd.springer.com/article/10.1007/BF01332495",
msc = "47B35, 47D25",
keywords = "?",
mathreviewsid = "?",
}
@ARTICLE { Vasilevski_1999_structure_poly_Bergman,
author = "Vasilevski, Nikolai L.",
title = "On the structure of {B}ergman and poly-{B}ergman spaces",
journal = "Integr. Equ. Oper. Theory",
fulljournal = "Integral Equations and Operator Theory",
publisher = "Birkh{\"{a}}user Basel",
issn = "0378-620X",
year = "1999",
volume = "33",
issue = "4",
pages = "471--488",
doi = "10.1007/BF01291838",
url = "http://www.springerlink.com/content/p6714x772m2k5637, http://rd.springer.com/article/10.1007/BF01291838",
msc = "46E20, 46E22",
keywords = "?",
mathreviewsid = "?",
}
@ARTICLE { Vasilevski_2003_commutative_algebras_hyperbolic_geometry,
author = "Vasilevski, Nikolai L.",
title = "{B}ergman space structure, commutative algebras of {T}oeplitz operators, and hyperbolic geometry",
journal = "Integr. Equ. Oper. Theory",
fulljournal = "Integral Equations and Operator Theory",
publisher = "Birkh{\"{a}}user Basel",
issn = "0378-620X(p) 1420-8989(e)",
year = "2003",
volume = "46",
issue = "2",
pages = "235--251",
url = "http://www.springerlink.com/content/11cj9rv1p0f5764j, http://rd.springer.com/article/10.1007/s000200300026",
doi = "10.1007/s000200300026",
mathreviewsid = "?",
msc2000 = "Primary 47B35; Secondary 47E20",
keywords = "Toeplitz operators, Bergman space, hyperbolic geometry",
abstract = "We exhibit a surprising but natural connection among the Bergman space structure,
commutative algebras of Toeplitz operators and pencils of hyperbolic straight lines.
The commutative $C*$-algebras of Toeplitz operators on the unit disk can be classified as follows.
Each pencil of hyperbolic straight lines determines the set of symbols consisting of functions
which are constant on corresponding cycles, the orthogonal trajectories to lines forming a pencil.
It turns out that the $C*$-algebra generated by Toeplitz operators with this class of symbols is commutative.",
}
@BOOK { Vasilevski_2008_book_Toeplitz_Bergman,
author = "Vasilevski, Nikolai L.",
title = "Commutative Algebras of {T}oeplitz Operators on the {B}ergman Space",
publisher = "Birkh{\"{a}}user",
year = "2008",
series = "Operator Theory: Advances and Applications",
volume = "185",
pages = "417",
address = "Basel--Boston--Berlin",
isbn = "978-3-7643-8725-9",
url = "http://www.springer.com/birkhauser/mathematics/book/978-3-7643-8725-9",
msc = "30C40, 46E22, 47A25, 47B10, 47B35, 47C15, 47L15, 81S10",
keywords = "?",
mathreviewsid = "?",
}
@BOOK { Widder_1946_Laplace_transform,
author = "Widder, David Vernon",
title = "The {L}aplace transform",
isbn = "",
edition = "",
publisher = "",
year = "1946",
series = "",
volume = "",
msc = "",
keywords = ""
mathreviewsid = "",
abstract = "",
}
@ARTICLE { Zhou_Dong_2009_algebraic_properties_Toeplitz_radial_unit_disc,
author = "Zhou, Ze-Hua and Dong, Xing-Tang",
title = "Algebraic properties of {T}oeplitz Operators with radial symbols
on the {B}ergman space of the unit ball",
journal = "Integr. equ. oper. theory",
fulljournale = "Integral Equations and Operator Theory",
issn = "0378-620X",
publisher = "Birkh{\"{a}}user Verlag Basel/Switzerland",
year = "2009",
volume = "64",
issue = "",
pages = "137--154",
doi = "10.1007/s00020-009-1677-y",
url = "",
msc2000 = "Primary 47B35; Secondary 32A36",
keywords = "Toeplitz operator, Bergman space, Mellin transform, radial symbol, quasihomogeneous symbol",
mathreviewsid = "",
abstract =
"In this paper, we discuss some algebraic properties
of Toeplitz operators with radial symbols on the Bergman space of the unit ball in $\mathbb{C}^n$.
We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator.
Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols.
Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form
$\xi^k\varphi$ is studied, where $k\in\mathbb{Z}^n$ and $\varphi$ is a radial function.",
}
@ARTICLE { Zhou_Chen_Dong_2011_Berezin_radial_unit_ball,
author = "Zhou, Ze-Hua and Chen, Wei-Li and Dong, Xing-Tang",
affiliation = "Department of Mathematics, Tianjin University, Tianjin, 300072 People’s Republic of China",
title = "The {B}erezin transform and radial operators on the {B}ergman space of the unit ball",
journal = "Complex Analysis and Operator Theory",
issn = "1661-8254",
publisher = "Birkh{\"{a}}user Basel",
year = "2011",
volume = "",
issue = "",
pages = "1--17",
doi = "10.1007/s11785-011-0145-2",
url = "http://www.springerlink.com/content/v331157021t517mv",
msc2000 = "Primary 47B35; Secondary 47B38, 46E15, 32A37",
keywords = "Toeplitz operators, Bergman space, radial operator, Berezin transform",
mathreviewsid = "",
abstract =
"In this paper, we investigate the connection between compactness of operators on the Bergman space
and the boundary behaviour of the corresponding Berezin transform.
We prove that for a class of operators that we call radial operators,
an oscillation criterion and diagonal are sufficient conditions under which the compactness of an operator is equivalent
to the vanishing of the Berezin transform on the unit sphere.
We further study a special class of radial operators, i.e.,
Toeplitz operators with a radial $L^1(B_n)$ symbol.",
}
@ARTICLE { Zorboska_2003_Berezin_radial,
author = "Zorboska, Nina",
title = "The {B}erezin transform and radial operators",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939(p), 1088-6826(e)",
year = "2003",
posted = "2002-07-02",
volume = "131",
number = "3",
pages = "793--800",
doi = "10.1090/S0002-9939-02-06691-1",
url = "http://www.ams.org/journals/proc/2003-131-03/S0002-9939-02-06691-1/home.html",
msc = "Primary 47B37, 47B38, 47B10; Secondary 30D55, 30H05, 47B35",
keywords = "Berezin transform, radial operator, Toeplitz operator",
mathreviewsid = "1937440 (2003h:47064)",
abstract =
"We analyze the connection between compactness of operators on the Bergman space
and the boundary behaviour of the corresponding Berezin transform.
We prove that for a special class of operators that we call radial operators,
an oscilation criterion is a sufficient condition under which the compactness of an operator
is equivalent to the vanishing of the Berezin transform on the unit circle.
We further study a special class of radial operators, i.e.,
Toeplitz operators with a radial $L^{1}(\mathbb{D})$ symbol.",
}
@ARTICLE { Zhu_1987_VMO_ESV_Toeplitz_Bergman,
author = "Zhu, Ke He",
title = "{VMO}, {ESV}, and {T}oeplitz operators on the {B}ergman space",
journal = "Trans. Amer. Math. Soc.",
publisher = "American Mathematical Society",
issn = "0002-9947(p), 1088-6850(e)",
year = "1987",
volume = "302",
pages = "617--646",
msc = "Primary 47B35; Secondary 30H05, 46L99",
mathreviewsid = "891638",
abstract =
"This paper studies the largest ${C^*}$-subalgebra $Q$ of ${L^\infty }({\mathbf{D}})$
such that the Toeplitz operators ${T_f}$ on the Bergman space $L_a^2({\mathbf{D}})$ with symbols $f$ in $Q$
have a symbol calculus modulo the compact operators.
$Q$ is characterized by a condition of vanishing mean oscillation near the boundary.
I also give several other necessary and sufficient conditions for a bounded function to be in $Q$.
After decomposing $Q$ in a ``nice'' way, I study the Fredholm theory of Toeplitz operators with symbols in $Q$.
The essential spectrum of ${T_f}(f \in Q)$ is shown to be connected and computable in terms of the Stone-C{\v{e}}ch compactification of ${\mathbf{D}}$.
The results in this article partially answer a question posed in [3] and give several new necessary and sufficient conditions
for a bounded analytic function on the open unit disc to be in the little Bloch space ${\mathcal{B}_0}$.",
}
@BOOK { Zhu_2004_book_spaces_holomorphic_unit_ball,
author = "Zhu, Kehe",
title = "Spaces of Holomorphic Functions on the Unit Ball",
isbn = "0-387-22036-4",
edition = "",
publisher = "Springer",
year = "2004",
series = "Graduate Texts in Mathematics",
volume = "226",
msc = "MSCM12198, SCM12198, SCM12007",
keywords = "",
mathreviewsid = "",
abstract = "",
}
@BOOK { Zhu_2007_book_operator_theory_function_spaces,
author = "Zhu, Kehe",
title = "Operator Theory in Function Spaces",
edition = "Second",
publisher = "American Mathematical Society",
year = "2007",
series = "Mathematical surveys and monographs",
volume = "138",
pages = "348",
isbn = "978-0-8218-3965-2",
url = "http://www.ams.org/bookpages/surv-138",
msc = "?",
keywords = "?",
mathreviewsid = "?",
abstract = "?",
}