Research output: Chapter in Book/Report/Conference proceeding › Chapter › Research › peer-review
Toeplitz operators via sesquilinear forms. / Rozenblum, Grigori; Vasilevski, Nikolai.
Operator Theory: Advances and Applications. Springer Nature, 2018. p. 287-304 (Operator Theory: Advances and Applications; Vol. 262).Research output: Chapter in Book/Report/Conference proceeding › Chapter › Research › peer-review
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TY - CHAP
T1 - Toeplitz operators via sesquilinear forms
AU - Rozenblum, Grigori
AU - Vasilevski, Nikolai
PY - 2018/1/1
Y1 - 2018/1/1
N2 - The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain common framework, based upon the extensive use of the language of sesquilinear form, for definition of Toeplitz operators for a ‘maximally wide’ class of ‘highly singular’ symbols. Besides covering all previously considered cases, such an approach permits us to introduce a further substantial extension of the class of admissible symbols that generate bounded Toeplitz operators. Although our approach is unified for all reproducing kernel Hilbert spaces, concrete operator consideration are given for Toeplitz operators acting on the standard Fock space, on the standard Bergman space on the unit disk (two leading examples in the classical theory of Toeplitz operators), and on the so-called Herglotz space consisting of the solutions of the Helmholtz equation.
AB - The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain common framework, based upon the extensive use of the language of sesquilinear form, for definition of Toeplitz operators for a ‘maximally wide’ class of ‘highly singular’ symbols. Besides covering all previously considered cases, such an approach permits us to introduce a further substantial extension of the class of admissible symbols that generate bounded Toeplitz operators. Although our approach is unified for all reproducing kernel Hilbert spaces, concrete operator consideration are given for Toeplitz operators acting on the standard Fock space, on the standard Bergman space on the unit disk (two leading examples in the classical theory of Toeplitz operators), and on the so-called Herglotz space consisting of the solutions of the Helmholtz equation.
KW - Bergman space
KW - Fock space
KW - Herglotz space
KW - Reproducing kernel hilbert space
KW - Sesquilinear form
KW - Toeplitz operators
UR - http://www.scopus.com/inward/record.url?scp=85044968013&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-62527-0_9
DO - 10.1007/978-3-319-62527-0_9
M3 - Chapter
AN - SCOPUS:85044968013
T3 - Operator Theory: Advances and Applications
SP - 287
EP - 304
BT - Operator Theory
PB - Springer Nature
ER -
ID: 50650045