Toeplitz and Hankel operators on the Paley-Wiener space
Let \(I=[-\pi,\pi]\). For a function f defined on I we denote by \(\chi^ f\) the function on R which agrees with f on I and vanishes on \(R\setminus I\). Denote the Fourier transform of g by \({\mathcal F}g=\hat g\) and the inverse Fourier transform of f by \({\mathcal F}^{-1}f=f^{\vee}\). The Paley-Wiener space, PW, is defined to be the image of \(L^ 2(I)\) under \({\mathcal F}^{-1}\). We norm PW be requiring \({\mathcal F}\) to be an isometry. The Toeplitz operator with ymbol \(\phi\) is the operator \(T_{\phi}\) with domain PW given by \(T_{\phi}(f)=P(\phi f)\) where P denotes the orthogonal projection from \(L^ 2\) to PW. The Hankel operator with symbol \(\phi\) is the operator \(H_{\phi}\) defined by \(H_{\phi}(f)=P(\phi \bar f).\) For any z in C and any t in R set \(f_ z=f_ z(t)=\chi_{2I}(t)e^{izt}\). The author shows that finite sums \(\phi =\sum c_ if^{\vee}_{z_ i}\) give finite rank operators and those are essentially the only finite rank \(H_{\phi}\). When \(\phi^{\wedge}\) is supported in \(R_+\), \(T_{\phi}\) has been studied by \textit{D. Sarason} [Trans. Am. Math. Soc. 127, 179-203 (1967; Zbl 0145.393)] and others. Pick and fix \(\psi_ L\) in \(C_ 0^{\infty}\) which has \(\sup p(\psi_ L)\subset [-4\pi,-\pi /2]\), and \(\psi_ L(x)=1\) for x in [- 3\(\pi\),-\(\pi\) ]. Set \(\psi_ R(x)=\psi_ L(-x)\) and define \(\psi_ c\) to be off 2I and \(\psi_ c=1-\psi_ R-\psi_ L\) on I. Define \(\phi_ L\) by \(\phi_ L=(\psi_ L\phi^{\wedge})^{\vee}\) and set \(T_ L=T_{\phi_ L}\). Let \(P_+\) be the orthogonal projection from \(L^ 2\) to the associated Hardy space \(H^ 2\), \(P_-=I-P_+\) and \(\theta =e^{i\pi x}\). The author shows that \(\| T_{\phi}\| \cong \| P_-({\bar \theta}^ 2\phi_ R)\|_*+\| P_+(\theta^ 2\phi_ L)\|_*+\| \phi_ c\|_{\infty}\) where \(\|\|_*\) denotes BMO norm. He gives characterizations of the symboles of \(T_{\phi}\) in the compact ideal or the Schatten-von Neumann ideals. Moreover he studies the commutator of multiplication operators with the discrete Hilbert transform. In the last section, many questions are raised.
commutator of multiplication operators with the discrete Hilbert transform
scientific article; zbMATH DE number 4032430
scientific article