Rend. Circ. Mat. Palermo, II. Ser https://doi.org/10.1007/s12215-018-0353-y On a theorem of Bishop and commutants of Toeplitz operators in C 1 1 Sönmez Sahuto ¸ glu ˘ · Akaki Tikaradze Received: 23 August 2017 / Accepted: 19 May 2018 © Springer-Verlag Italia S.r.l., part of Springer Nature 2018 Abstract We prove an approximation theorem on a class of domains in C on which the ∂-problem is solvable in L . Furthermore, as a corollary, we obtain a version of the Axler– Cuck ˇ ovic–Rao ´ theorem in higher dimensions. Keywords Bishop’s theorem · Pseudoconvex domain · Toeplitz operator Mathematics Subject Classiﬁcation Primary 46J15; Secondary 32A65 n ∞ Let be a domain in C and φ be a complex-valued function on .Let H () and H ()[φ] denote the set of bounded holomorphic functions on and the algebra gen- erated by φ over H (), respectively. In 1989, Christopher Bishop proved the following approximation theorem (see [6, Theorem 1.2]). Theorem (Bishop) Let be an open set in C and f be a bounded holomorphic function on that is non-constant on every connected component of .Then H ()[ f ] is dense in C () in the uniform
Rendiconti del Circolo Matematico di Palermo – Springer Journals
Published: Jun 2, 2018
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