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Commutators and semicommutators of Toeplitz operators

机译：Toeplitz算子的交换子和半交换器

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Let phi and psi be two essentially bounded functions on the unit circle T. This paper is devoted to the study of the semicommutator T(sub phi-psi) - T(sub phi)T(sub psi) of the Toeplitz operators T(sub phi) and T(sub psi). We prove that if phi is inner and P(sub +) (psi-phi-bar) is an element of B(sup 1/p)(sub p) then the semicommutator T(sub phi-psi) - T(sub phi)T(sub psi) is in the Schatten-Von-Neumann class sigma(sub p), p > 0. Here B(sup 1/p)(sub p) denote the Besov class, P(sub +) is the orthogonal projection from L(sup 2) onto the Hardy space H(sup 2). Moreover if phi is also continuous, then T(sub phi-psi) - T(sub phi)T(sub psi) is of finite rank. An example based on the Hardy-Littlewood series shows that the result fails if we suppose phi only continuous. We also give some sufficient conditions on the Fourier coefficients of the symbols phi and psi which implies that the commutator (T(sub phi), T(sub psi)) belongs to sigma(sub p), p > 0. (author). 14 refs. (Atomindex citation 28:009450)

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