The functional analytic approach establishes the Corona theorem in the theory of SCV
Abstract
We give an affirmative solution of the original Corona problem (1962) for certain bounded pseudoconvex domains or polydomains in ℂn, including the balls and polydiscs, by using the Gleason solvability property of such domains for H∞(G). In fact, we establish an equivalency of the Corona problem and Gleason's problem (1964) in the theory of several complex variables. Indeed, we extend our recent works on both problems based on the functional analytic approaches, and extend recent work of Clos, as well as apply the usual Banach algebra method. As an application, we establish the I. J. Schark's theorem for these G's. We also obtain a welcome extension of a result of Krantz and Li (1995) for a bounded, strictly pseudoconvex domain in ℂn with C3 boundary.
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BibTeX
@misc{patel2026functional,
title = {{The functional analytic approach establishes the Corona theorem in the theory of SCV}},
author = {S. R. Patel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39425},
eprint = {2610.39425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39425},
}
RIS
TY - UNPB
TI - The functional analytic approach establishes the Corona theorem in the theory of SCV
AU - S. R. Patel
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39425
UR - https://arxiv.org/abs/2610.39425
AB - We give an affirmative solution of the original Corona problem (1962) for certain bounded pseudoconvex domains or polydomains in ℂ^n, including the balls and polydiscs, by using the Gleason solvability property of such domains for H^∞(G). In fact, we establish an equivalency of the Corona problem and Gleason's problem (1964) in the theory of several complex variables. Indeed, we extend our recent works on both problems based on the functional analytic approaches, and extend recent work of Clos, as well as apply the usual Banach algebra method. As an application, we establish the I. J. Schark's theorem for these G's. We also obtain a welcome extension of a result of Krantz and Li (1995) for a bounded, strictly pseudoconvex domain in ℂ^n with C^3 boundary.
ER -