Toeplitz and Wiener-Hopf operators in 𝐻^{∞}+𝐶

Abstract

[lO] for earlier results) that <r(7^) consists of the range of 0 on T along with those X for which the index of X with respect to the curve determined by <f> is different from zero. In this note we describe <r(T<t) for 4> in the linear span of iJ 00 and C. (This manifold is actually a closed subalgebra of Z, 00 .) We show that such a T<f, is invertible if its harmonic extension $ to D is bounded away from zero on a neighborhood of T and the index of the curve $(Re id ) is zero for R sufficiently large. Our technique can be viewed as an extension of that used in [ó] to determine <r{T<i) for <j> in C.In §3 we indicate how to extend our results to determine the index of a certain class of vector-valued Toeplitz operators (or systems of Toeplitz operators). In this we generalize certain of the results of Gohberg and Kreïn in [7]. We conclude by describing how our results can be applied to the study of Wiener-Hopf operators both in the scalar case and the vector-valued case using the isomorphism exhibited in [5].We only outline our proofs and complete details will appear elsewhere.2, We begin by recalling some facts about Fredholm operators. Let £ denote the algebra of bounded operators on H 2 , JC the uniformly closed two-sided ideal of compact operators in £, and w the homomorphism of £ onto <£/3C. An operator A in £ is said to be a Fredholm operator if A has a closed range and both a finite dimen-895

Sums of invariant subspaces

“…Clearly supp φ is a closed subset. Now Douglas [3] has shown that a function in H°° 4-C is invertible if and only if its harmonic extension is bounded away from 0 in some annulus r < | z | < 1. Thus a unimodular function φ in H°° + C is invertible if and only if suppφ is empty.…”

One sided invertibility of matrices over commutative rings, corona problems, and Toeplitz operators with matrix symbols

“…[2]). Finally, expanding det Φ e along the last column, we obtain 14) which is equal to the (n, n) entry of the product (adj Φ e ) · (adj Ψ e ), where we denote by adj X ∈ A n×n the algebraic adjoint (adjugate) of a matrix X ∈ A n×n . Since Ψ e and Φ e are inverses of each other, then so are adj Φ e and adj Ψ e , and (2.14) is equal to 1, as claimed.…”

Algebras of functions on the unit circle

“…By Theorem 1 there is a smallest such algebra, namely, the closed subalgebra of L 00 generated by H 00 and C. Somewhat unexpectedly, this algebra turns out to equal H 00 + C, the linear hull of H 00 and C (a fact which seems to have first been pointed out in [20]). The algebra H 00 + C arises, among other ways, in the study of Toeplitz operators [5], [8] and in a problem in prediction theory investigated by H. Helson and the author [14], [21]. In the present section I shall describe a few of the basic properties of tf 00 + C. The key observation needed to prove that H 00 + C is closed in L 00 is this: (*) If f is any function in C, then dist(/, A) = dist( ƒ, H °°).…”

“…Such a criterion for H 00 + C has been established and exploited by R. G. Douglas [5] ; it involves the behavior inside the unit disk of the harmonic extensions of the functions in H 00 + C. We let P denote the Poisson kernel : P(r, 0) = (1 -r 2 )/(l -2r cos 9 + r 2 ). The theorem is an easy consequence of the following lemma.…”