Absolutely summing multipliers on \(H^p\) spaces
A linear operator \(T\) from a normed linear space \(X\) to a normed linear space \(Y\) is \(r\)--summing if for all natural numbers \(N\) and all \(x_1,...,x_N\in X\), there exists an absolute constant \(K>0\) such that \[ \left(\sum_{n=1}^N \|Tx_n\|^r\right)^{1/r} \leq K\sup\left\{\left(\sum_{n=1}^N|\phi(x_n)|^r\right)^{1/r}: \phi\in X^{\ast}, \|\phi\|\leq 1\right\}. \] The infimum of all such \(K\) is the \(r\)--summing norm of \(T\). For a sequence \( a=(a_n)\) of complex numbers, the multiplier operator \(d_a\) of the Hardy space \(H^p \) into the sequence space \(\ell_q \) is the map defined by \(d_af = (\hat f(n)a_n)_{n=1}^{\infty}\), where \((\hat f(n))\) is the sequence of Fourier coefficients of \(f\in H^p\). In the paper the author obtains a complete characterization for \(d_a\) to be \(2\)--summing or \(1\)--summing when \(1\leq p\leq 2\) and \(1\leq q \leq \infty\), and gives upper and lower estimates for the \(2\)--summing and \(1\)--summing norms in the case \(2<p\leq \infty\) and \(1\leq q \leq \infty\).
scientific article; zbMATH DE number 1346372
scientific article