Details
| Original language | English |
|---|---|
| Pages (from-to) | 223-240 |
| Number of pages | 18 |
| Journal | Integral Equations and Operator Theory |
| Volume | 72 |
| Issue number | 2 |
| Publication status | Published - 1 Dec 2011 |
| Externally published | Yes |
Abstract
Studying commutative C*-algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C*-algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each standard weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent and quasi-nilpotent. Recently it was observed in Vasilevski (Integr Equ Oper Theory. 66:141-152, 2010) that there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach, and being extended to C*-algebras they became non-commutative. These results were extended then to the classes of symbols, subordinated to the quasi-hyperbolic and quasi-parabolic groups. In this paper we prove the analogous commutativity result for Toeplitz operators whose symbols are subordinated to the quasi-nilpotent group. At the same time we conjecture that apart from the known C*-algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group and which are commutative on each weighted Bergman space.
Keywords
- Commutative Banach algebra, Quasi-homogeneous, Quasi-nilpotent, Toeplitz operator, Weighted Bergman space
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Algebra and Number Theory
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In: Integral Equations and Operator Theory, Vol. 72, No. 2, 01.12.2011, p. 223-240.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action
AU - Bauer, Wolfram
AU - Vasilevski, Nikolai
N1 - Funding Information: W. Bauer has been supported by an “Emmy-Noether scholarship” of Forschungsgemeinschaft). N. Vasilevski has been partially supported Project 102800, México. Copyright: Copyright 2012 Elsevier B.V., All rights reserved.
PY - 2011/12/1
Y1 - 2011/12/1
N2 - Studying commutative C*-algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C*-algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each standard weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent and quasi-nilpotent. Recently it was observed in Vasilevski (Integr Equ Oper Theory. 66:141-152, 2010) that there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach, and being extended to C*-algebras they became non-commutative. These results were extended then to the classes of symbols, subordinated to the quasi-hyperbolic and quasi-parabolic groups. In this paper we prove the analogous commutativity result for Toeplitz operators whose symbols are subordinated to the quasi-nilpotent group. At the same time we conjecture that apart from the known C*-algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group and which are commutative on each weighted Bergman space.
AB - Studying commutative C*-algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C*-algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each standard weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent and quasi-nilpotent. Recently it was observed in Vasilevski (Integr Equ Oper Theory. 66:141-152, 2010) that there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach, and being extended to C*-algebras they became non-commutative. These results were extended then to the classes of symbols, subordinated to the quasi-hyperbolic and quasi-parabolic groups. In this paper we prove the analogous commutativity result for Toeplitz operators whose symbols are subordinated to the quasi-nilpotent group. At the same time we conjecture that apart from the known C*-algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group and which are commutative on each weighted Bergman space.
KW - Commutative Banach algebra
KW - Quasi-homogeneous
KW - Quasi-nilpotent
KW - Toeplitz operator
KW - Weighted Bergman space
UR - http://www.scopus.com/inward/record.url?scp=84855651342&partnerID=8YFLogxK
U2 - 10.1007/s00020-011-1927-7
DO - 10.1007/s00020-011-1927-7
M3 - Article
AN - SCOPUS:84855651342
VL - 72
SP - 223
EP - 240
JO - Integral Equations and Operator Theory
JF - Integral Equations and Operator Theory
SN - 0378-620X
IS - 2
ER -