Zero products of Toeplitz operators. (English) Zbl 1170.47013

Let \(H^p\), \(1<p<\infty\), be the Hardy space of the unit circle \({\mathbb T}\) and \(H_m^p\) be the space of all vector-valued functions \(f=(f^1,\dots,f^m)\), where each \(f^j\in H^p\). Further, let \(L^\infty({\mathbb T },M_{m\times m}({\mathbb C}))\) denote the set of all \(m\times m\) matrices with essentially bounded entries. By \(T_u\) denote the Toeplitz operator on \(H^p_m\) with symbol \(u\in L^\infty({\mathbb T},M_{m\times m}({\mathbb C}))\). The main result of the paper says that if \(u_1,\dots,u_n\in L^\infty({\mathbb T},M_{m\times m}({\mathbb C}))\) and \(T_{u_1}\cdots T_{u_n}=0\), then \(\det u_k=0\) almost everywhere on \({\mathbb T}\) for some \(k\in\{1,\dots,n\}\).


47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30H05 Spaces of bounded analytic functions of one complex variable
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