TY - JOUR
T1 - Hermitian Weighted Composition Operators and Bergman Extremal Functions
AU - Cowen, Carl C.
AU - Gunatillake, Gajath
AU - Ko, Eungil
N1 - Funding Information:
E. Ko is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0087565). The authors would like to thank Ehwa Women’s University, Seoul, and Purdue University, West Lafayette, for their generosity in hosting the authors during their collaboration.
PY - 2013/2
Y1 - 2013/2
N2 - Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are (1 - w̄z)-κ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.
AB - Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are (1 - w̄z)-κ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.
KW - Bergman inner function
KW - Composition operator
KW - Hermitian operator
KW - Normal operator
KW - Projected kernel function
KW - Spectral measure
KW - Weighted Bergman space
KW - Weighted Hardy space
KW - Weighted composition operator
UR - http://www.scopus.com/inward/record.url?scp=84872874114&partnerID=8YFLogxK
U2 - 10.1007/s11785-011-0185-7
DO - 10.1007/s11785-011-0185-7
M3 - Article
AN - SCOPUS:84872874114
SN - 1661-8254
VL - 7
SP - 69
EP - 99
JO - Complex Analysis and Operator Theory
JF - Complex Analysis and Operator Theory
IS - 1
ER -