Closed Range Type Properties of Toeplitz Operators on the Bergman Space and the Berezin Transform
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We characterize the multiplication operators with closed range on the Bergman space in terms of the Berezin transform, and apply this characterization to finite products of interpolating Blaschke products. We give some necessary and some sufficient conditions for invertibility of general Toeplitz operators on the Bergman space. We determine the Fredholm Toeplitz operators with \(BMO^1\) symbols and the invertible Toeplitz operators with nonnegative symbols, when their Berezin transform is bounded and of vanishing oscillation.
KeywordsMultiplication operator Toeplitz operator Bergman space Berezin transform Closed range operator Invertible operator Fredholm operator Interpolating Blaschke product
Mathematics Subject Classification47B35 47A53 30H20 30H35 30J10
Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada (CA) (Grant No. 1304332).