Details
| Original language | English |
|---|---|
| Pages (from-to) | 213-235 |
| Number of pages | 23 |
| Journal | Journal of Mathematical Analysis and Applications |
| Volume | 386 |
| Issue number | 1 |
| Publication status | Published - 29 Jul 2011 |
| Externally published | Yes |
Abstract
Let T=Tzlz̄k with l,k∈N{double-struck}0 be a Toeplitz operator with monomial symbol acting on the Segal-Bargmann space over the complex plane. We determine the symbols ψ of polynomial growth at infinity such that Tψ and Tzlz̄k commute on the space of all holomorphic polynomials. By using polar coordinates we represent ψ as an infinite sum ψ(reiθ)=∑j=-∞-∞ψj(r)eijθ. Then we are able to reduce the above problem to the case of quasi-homogeneous symbols ψ=ψjeijθ. We obtain the radial part ψj(r) in terms of the inverse Mellin transform of an expression which is a product of Gamma functions and a trigonometric polynomial. If we allow operator symbols of higher growth at infinity, we point out that in some of the cases more than one Toeplitz operator Tψjeijθ exists commuting with T.
Keywords
- Mellin transform, Monomial symbols, Symbols with polynomial growth
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
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In: Journal of Mathematical Analysis and Applications, Vol. 386, No. 1, 29.07.2011, p. 213-235.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Commuting Toeplitz operators with quasi-homogeneous symbols on the Segal-Bargmann space
AU - Bauer, Wolfram
AU - Issa, Hassan
N1 - Funding Information: E-mail addresses: wbauer@uni-math.gwdg.de (W. Bauer), hissa@uni-math.gwdg.de, baladalamin@hotmail.com (H. Issa). 1 The authors have been supported by an “Emmy-Noether scholarship” of DFG (Deutsche Forschungsgemeinschaft). Copyright: Copyright 2011 Elsevier B.V., All rights reserved.
PY - 2011/7/29
Y1 - 2011/7/29
N2 - Let T=Tzlz̄k with l,k∈N{double-struck}0 be a Toeplitz operator with monomial symbol acting on the Segal-Bargmann space over the complex plane. We determine the symbols ψ of polynomial growth at infinity such that Tψ and Tzlz̄k commute on the space of all holomorphic polynomials. By using polar coordinates we represent ψ as an infinite sum ψ(reiθ)=∑j=-∞-∞ψj(r)eijθ. Then we are able to reduce the above problem to the case of quasi-homogeneous symbols ψ=ψjeijθ. We obtain the radial part ψj(r) in terms of the inverse Mellin transform of an expression which is a product of Gamma functions and a trigonometric polynomial. If we allow operator symbols of higher growth at infinity, we point out that in some of the cases more than one Toeplitz operator Tψjeijθ exists commuting with T.
AB - Let T=Tzlz̄k with l,k∈N{double-struck}0 be a Toeplitz operator with monomial symbol acting on the Segal-Bargmann space over the complex plane. We determine the symbols ψ of polynomial growth at infinity such that Tψ and Tzlz̄k commute on the space of all holomorphic polynomials. By using polar coordinates we represent ψ as an infinite sum ψ(reiθ)=∑j=-∞-∞ψj(r)eijθ. Then we are able to reduce the above problem to the case of quasi-homogeneous symbols ψ=ψjeijθ. We obtain the radial part ψj(r) in terms of the inverse Mellin transform of an expression which is a product of Gamma functions and a trigonometric polynomial. If we allow operator symbols of higher growth at infinity, we point out that in some of the cases more than one Toeplitz operator Tψjeijθ exists commuting with T.
KW - Mellin transform
KW - Monomial symbols
KW - Symbols with polynomial growth
UR - http://www.scopus.com/inward/record.url?scp=80052824901&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2011.07.058
DO - 10.1016/j.jmaa.2011.07.058
M3 - Article
AN - SCOPUS:80052824901
VL - 386
SP - 213
EP - 235
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
ER -