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(Weak) compactness of Hankel operators on \(BMOA\). (English) Zbl 1303.47042

Let \(H^p=H^p(\mathbf T)\), \(1\leq p<\infty\), be the Hardy space over the unit circle \(\mathbf T\) which consists of all functions \(f\) in \(L^p=L^p(\mathbf T)\) such that the Fourier coefficients \(\hat f(n)\) all vanish for \(n<0\). Let \(BMOA\) be the dual of \(H^1\), which is identified with \(BMO\cap H^1\), where \(BMO=BMO(\mathbf T)\) is the standard space of functions with bounded mean oscillations over \(\mathbf T\). For \(a\in H^2\) with \(\hat a(0)=0\), the Hankel operator \(H_a\) with symbol \(a\) is defined by \(H_a(f)=P(a Jf)\), where \(Jf(\zeta)=\overline{\zeta}f(\overline{\zeta})\) and \(P: L^2 \to H^2\) is the Hilbert space orthogonal projection, i.e., the Riesz projection.
When \(H_a\) is bounded on \(BMOA\), the author obtains several equivalent conditions for the (weak) compactness of \(H_a\) on \(BMOA\). In particular, it is shown that the compactness and the weak compactness of \(H_a\) on \(BMOA\) are the same.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30H35 BMO-spaces
30H10 Hardy spaces
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