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Hyponormal Toeplitz operators with rational symbols. (English) Zbl 1122.47023

Summary: In this paper, we consider the self-commutators of Toeplitz operators \(T_{\varphi}\) with rational symbols \(\varphi\) using the classical Hermite–Fejér interpolation problem. Our main theorem is as follows. Let \(\varphi=\overline{g}+f\in L^{\infty}\) and let \(f=\theta\overline{a}\) and \(g=\theta\overline{b}\), where \(\theta\) is a finite Blaschke product of degree \(d\) and \(a,b\in \mathcal H(\theta):=H^2\ominus \theta H^2\). Then \(\mathcal H (\theta)\) is a reducing subspace of \([T_{\varphi}^*,T_{\varphi}]\), and \([T_{\varphi}^*,T_{\varphi}]\) has the following representation relative to the direct sum \(\mathcal{H}(\theta)\oplus \mathcal{H}(\theta)^\perp\): \[ [T_{\varphi}^*,T_{\varphi}]=A(a)^* W M(\varphi) W^*A(a)\oplus 0_\infty, \] where \(A(a):=P_{\mathcal{H}(\theta)}M_a|_{\mathcal{H}(\theta)}\) (\(M_a\) is the multiplication operator with symbol \(a\)), \(W\) is the unitary operator from \(\mathbb{C}^d\) onto \(\mathcal{H}(\theta)\) defined by \(W:=(\phi_1,\ldots,\phi_d)\) (\(\{\phi_j\}\) is an orthonormal basis for \(\mathcal{H}(\theta)\)), and \(M(\varphi)\) is a matrix associated with the classical Hermite–Fejér interpolation problem. Hence, in particular, \(T_\varphi\) is hyponormal if and only if \(M(\varphi)\) is positive. Moreover, the rank of the self-commutator \([T_\varphi^*,T_\varphi]\) is given by \(\text{rank}\,[T_\varphi^*,T_\varphi]=\text{rank}\,M(\varphi)\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B20 Subnormal operators, hyponormal operators, etc.
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