Operator properties of random rearrangements (Q1892808)

Let \(X= (x_{ij})\) be an \(n\times n\) matrix, \(\sigma_n\) be the permutation group on \(\{1,\dots, n\}\), \(\{s_k\}\) the decreasing rearrangement of \((x_{ij})\), \(Q\) a one-to-one mapping of \(\sigma_n\) onto \(\{1,\dots, n!\},\) \(1\leq q\leq \infty\). The paper deals with properties of the operator \(T_q\) acting between \(n\times n\) matrices and the space of step functions, defined by \[ T_q X(t)= \Biggl( \sum^n_{i= 1} |x_{i, \pi(i)}|^q\Biggr)^{1/q}\qquad (Q\pi- 1)/n!\leq Q\pi/n! \] for \(\pi\in \sigma_n\). Put \(SX(t)= s_k\) for \((k- 1)/n\leq t< k/n\), \(1\leq k\leq n\). The author finds quantities (independent of \(Q\)) which are equivalent to \(|T_q X|_E\), where \(E\) is either a r.i. space and \(|T_q Y|_E\leq c|SY|_E\) for every diagonal matrix \(Y\), or a Lorentz space \(\Lambda(\varphi)\), or \(E\) is a r.i. space containing some \(L_p\) \((1< p< \infty)\). This is an extension of a previous author's paper [Russ. J. Math. Phys. 1, 403-405 (1993)] and a paper of \textit{S. Kwapién} and \textit{C. Schütt} in [Stud. Math. 82, 81-106 (1985; Zbl 0579.46013)].