Abstract
This paper studies the largest C ∗ {C^*} -subalgebra Q Q of L ∞ ( D ) {L^\infty }({\mathbf {D}}) such that the Toeplitz operators T f {T_f} on the Bergman space L a 2 ( D ) L_a^2({\mathbf {D}}) with symbols f f in Q Q have a symbol calculus modulo the compact operators. Q 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 Q . After decomposing Q Q in a "nice" way, I study the Fredholm theory of Toeplitz operators with symbols in Q Q . The essential spectrum of T f ( f ∈ Q ) {T_f}(f \in Q) is shown to be connected and computable in terms of the Stone-Cěch compactification of D {\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 B 0 {\mathcal {B}_0} .
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
