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Fredholm Toeplitz Operators

1970; American Mathematical Society; Volume: 26; Issue: 1 Linguagem: Inglês

10.2307/2036815

1088-6826

R. G. Douglas, Donald Sarason,

Spectral Theory in Mathematical Physics

Whether a Toeplitz operator on H2 is Fredholm is shown to depend only on the local behavior of the inducing function.Let L2 and L°° denote the Lebesgue spaces of square integrable and essentially bounded functions with respect to normalized Lebesgue measure on the unit circle in the complex plane.Let H2 and H°° denote the corresponding Hardy spaces.For in Va, the Toeplitz operator induced by (f> is the operator T4, on H2 defined by T$f -P( f) ;here, P stands for the orthogonal projection in L2 with range H2.A great deal of effort has been devoted to finding criteria, in terms of the behavior of 4>, for the invertibility of T^,; a summary of most of the known results can be found in [2].All such criteria involve the global behavior of cj>.In contrast to this, we show here that whether or not T$ is a Fredholm operator depends, in a sense, only on the local behavior of .We recall that a Hubert space operator is said to be left-Fredholm if it is left-invertible modulo the compact operators, or, equivalently, if it has a closed range and a finite dimensional null space.An operator is said to be right-Fredholm if its adjoint is left-Fredholm, semi-Fredholm if it is either right-Fredholm or left-Fredholm, and Fredholm if it is both right-Fredholm and left-Fredholm.Theorem.Let be a function in L°°.Assume that for each point X on the unit circle there is an open subarc A of the unit circle containing X and a function f in L°° such that f\A = \A and T¡ is left-Fredholm.Then T$ is left-Fredholm.This theorem was suggested by, and contains as a corollary, a recent result of H. Widom and the first author [3].The proof uses the following three known results, which are due, respectively, to Hartman and Wintner, Coburn, and Rabindranathan.Lemma A [5].If fis in L°° and fis not essentially bounded away from zero, then Tf is not bounded below.