The invertibility of Toeplitz plus Hankel operators with subordinated operators of even index. (English) Zbl 1476.47024
Summary: Toeplitz plus Hankel operators \(T(a) + H(b)\) acting on Hardy spaces \(H^p(\mathbb{T}),\ p \in(1, \infty)\), where \(\mathbb{T}\) is the unit circle, are studied. If the functions \(a, b \in L_\infty(\mathbb{T})\) satisfy the relation \(a(t) a(1 / t) = b(t) b(1 / t)\), \(t \in \mathbb{T}\) and the Toeplitz operators \(T(a b^{- 1})\) and \(T(a \widetilde{b}^{- 1}),\ \widetilde{b}(t) = b(1 / t)\) have even indices, necessary and sufficient conditions for the invertibility of \(T(a) + H(b)\) are established and efficient formulas for their inverses are obtained. Moreover, it is shown that for any \(n \in \mathbb{N}\), there are invertible operators \(T(a) + H(b)\) such that \(\operatorname{ind} T(ab^{-1}) = -2n\) and \(\operatorname{ind} T(a \widetilde{b}^{-1}) = 2n\).
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 47B48 | Linear operators on Banach algebras |
| 47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |
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