### Toeplitz Operators on H 2 Spaces

Allen Devinatz
1964 Transactions of the American Mathematical Society
1. Let L2 be the usual Lebesgue space with respect to Haar measure (normalized to one) on the unit circle and H2 those elements of L2 whose Fourier transforms vanish on the negative integers. The unitary map which takes any element of L2 into its Fourier transform in I2 (the space of square summable sequences on the integers) will take H2 unitarily onto /+ (the sequences of I2 whose values vanish on the negative integers). Let qb be a bounded measurable function on the circle group and P the
more » ... group and P the projection from L2 onto H2. By the Laurent operator L^ on L2 into itself we shall simply mean the operator defined by L^g = qbg. The Toeplitz operator T¿ shall be PL¿ restricted to H2. The unitary Parseval operator from L2 onto I2 will induce corresponding Laurent and Toeplitz operators on I2 and l\ respectively which, of course, will be operators defined by convolution equations. The space H2 is clearly the closure in L2 of the algebra A consisting of continuous functions on the circle group whose Fourier coefficients vanish on the negative integers. The algebra A is a prototype of what in the past few years has come to be called a Dirichlet algebra. As in the circle group case, a general Dirichlet algebra A can be used to obtain L2 and H2 spaces and Laurent and Toeplitz operators can be defined. We are interested in the question under what circumstances a Toeplitz operator is invertible; i.e., is a one-one map of H2 onto itself? We have solved this problem by showing that the problem of invertibility of a general Toeplitz operator can be reduced to the problem of invertibility of a special type of Toeplitz operator and that, in turn, this latter problem is equivalent to a problem solved by Helson and Szegö . The invertibility problem on the transform space of H2 of the circle group, i.e., the invertibility problem of Toeplitz operators on I2, is of course the problem of solving a discrete Wiener-Hopf type convolution equation. Moreover, by a simple transformation the solution of the problem on l\ leads to a solution of the usual H2 type of Wiener-Hopf equation on the real line. The problem of invertibility for I2., also using the ideas of , was solved in 1960 by H. Widom and announced by him, in part, in . Without knowledge of Widom's work, we subsequently rediscovered his results and also found that the methods we had used would work