Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

Abstract

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator Ha is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for Ha to be compact on H 1 . The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2 . Mathematics Subject Classification (2000). Primary 47B35; Secondary 30D50, 30D55, 47A53.

(Weak) compactness of Hankel operators on $\mathit{BMOA}$

“…General considerations show the equivalences between (1), (3) and (5), between (2) and (4) and that (2) implies (1). Also, Theorem 1.5 shows the equivalence between (2) or (4) and (6). Therefore, it remains to prove that (5) It is easy to show that…”

“…All the previous estimates of the terms A and B are contained in [6] and played a substantial role in the proofs of Theorems 1.4 and 1.5. We included them here for the sake of completeness.…”

Toeplitz operators with piecewise continuous symbols on the Hardy space $H^1$

“…When p = 1 or p = ∞, both operators S and P are unbounded. For a ∈ L ∞ , we define the Toeplitz operator T a on H p with symbol a by [7]. Much less is known about their spectral properties when p = 1, and in particular, the Fredholm properties of Toeplitz operators are only understood for certain continuous symbols and symbols in the Douglas algebra.…”

Bounded and compact Toeplitz+Hankel matrices

“…In particular, T (a) is bounded on H p if and only if a ∈ L ∞ . For the boundedness of Toeplitz operators on the other Hardy spaces H p with 0 < p ≤ 1, see [16,19,23].…”