Abstract.
We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator H a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H a to be compact on H 1. The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2.
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The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25 National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.
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Papadimitrakis, M., Virtanen, J.A. Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties. Integr. equ. oper. theory 61, 573–591 (2008). https://doi.org/10.1007/s00020-008-1609-2
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DOI: https://doi.org/10.1007/s00020-008-1609-2