Index Formulas for Toeplitz Operators, Approximate Identities, and the Wolf–Havin Theorem
This is a review of a development in the 1970s and 1980s that was concerned with index formulas for block Toeplitz operators with discontinuous symbols. We cite the classical results by Douglas and Sarason, outline Silbermann’s approach to index formulas via algebraization, and embark on the replacement of the Abel–Poisson means by arbitrary approximate identities. One question caused by this development was whether the Abel–Poisson means are the only approximative identities that are asymptotically multiplicative on (H∞,H∞), and the review closes with Wolf and Havin’s theorem, which gives an affirmative answer to this question.
KeywordsToeplitz operator index formula approximate identity asymptotic multiplicativity BMO VMO Viktor Havin
Mathematics Subject Classification (2010)Primary 47B35. Secondary 30H05 42A10 46J15
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