Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Isometrics mở rộng của các nhà khai thác và vấn đề của subspaces bất biến. | J. OPERATOR THEORY 6 1981 51-54 Copyright by INCREST 198Ỉ OPERATORS COMMUTING WITH TOEPLITZ AND HANKEL OPERATORS MODULO THE COMPACT OPERATORS MICHAEL J. HOFFMAN For a separable Hilbert space H let tf H be the ideal of compact operators in the algebra of all bounded linear operators on H. For the essential commutant is TeX H ST TS e In 1975 K. Davidson showed that the essential commutant of the algebra generated by all Toeplitz operators is exactly the set of compact perturbations of Toeplitz operators with quasi-continuous symbol 2 . Using this and D. Sarason s characterization of quasi-continuous functions as those essentially bounded functions with vanishing mean oscillation 9 s. c. Power has shown that adding the requirement that the operator have compact commutator with all Hankel operators imposes only the additional condition that the symbol have symmetric Fourier series 6 p. 56 . Thus the Hankel and Toeplitz operators together do not generate all operators. This paper presents a proof of Power s result using neither Davidson s theorem nor functions of vanishing mean oscillation. Let D be the open unit disk in the complex plane and L2 L2 dD dỡ 27t . Let p be the orthogonal projection of L2 onto the Hardy space H2 of functions in L2 whose Fourier coefficients of negative index are 0. Let u be the unitary operator on L2 given by U ek e_k where ek is the basis function ek eư e . Equivalently Uf é e-ii . For fe L dD Lm let and for V L let p ry JE yj Let Mf be the multiplication operator defined on L2 by Mfg fg- let Tf be the Toeplitz operator on H2 given by Tfg PMfg and let Hf be the Hankel operator on H2 defined by Hfg PUMfg. Let m V Mf fe V y V Tf fe V and Hf f e F . If the difference of two operators T and s is compact write s. We will use several subspace of L . Let 27 H2 n L and c C dD . Sarason 7 p. 191 has shown that 7 Cis a uniformly closed subalgebra of L . It is not a -algebra but the space QC H C n 77 C of quasi- 52 MICHAEL J. HOFFMAN continuous
