×

A description of Hankel operators of class \({\mathfrak S}_ p\) for \(p>0\), an investigation of the rate of rational approximation, and other applications. (English. Russian original) Zbl 0561.47022

Math. USSR, Sb. 50, 465-494 (1985); translation from Mat. Sb., Nov. Ser. 122(164), No. 4, 481-510 (1983).
Let \(\phi\) be a bounded function on \(T\) and \(M_{\phi}\) be a Hankel operator on \(H^ 2\) with symbol \(\phi\) defined by the equality: \(H_{\phi}f=p_-\phi f,\quad f\in H^ 2,\) and \(p_-\) be an orthogonal projection from \(L^ 2\) onto \(H^ 2_-=L^ 2\theta H^ 2.\) Let also \(B_ p^{\beta}\) be a Besov space and \({\mathfrak S}_ p\), \(0<p<\infty\), the class of Schatten-von Neumann. The main result is as follows.
Theorem 1. Let \(\phi \in L^{\infty}\), \(0<p<\infty\). Then \(H_{\phi}\in {\mathfrak S}_ p\) if and only if \(p_{-\phi}\in B_ p^{1/p}.\)
In the case \(1\leq p<\infty\) this result was earlier obtained by the author [Mat. Sb., Nov. Ser. 113(155), 538–581 (1980; Zbl 0458.47022)]. In the case \(1<p<\infty\) see also [J. Peetre and E. Svensson, Math. Scand. 54, 221–241 (1984; Zbl 0535.42001)]; for \(p=1\) [R. R. Coifman and R. Rochberg, Astérisque, 77, 67–151 (1980; Zbl 0472.46040)].
The author considers also more general operators \(\Gamma_{\phi}^{\alpha,\beta}\), \(\alpha\),\(\beta\in R\), of the form \(\Gamma_{\phi}^{\alpha,\beta}=\{{\hat \phi}(n+k)(1+n)^{\alpha}(1+k)^{\beta}\}_{n,k\geq 0}\).
Theorem 2. Let \(0<p<1\), \(\alpha >-\), \(\beta >-\) and \(\phi\) be analytic in the unit disk. Then \(\Gamma_{\phi}^{\alpha,\beta}\in {\mathfrak S}_ p\) if and only if \(\phi \in B_ p^{1/p+\alpha +\beta}.\)
An application to the problem of rational approximation is also given.
Reviewer: N.K.Karapetianc

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
41A20 Approximation by rational functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47Gxx Integral, integro-differential, and pseudodifferential operators
PDFBibTeX XMLCite
Full Text: DOI EuDML