We study Y'(+), of all a is an element of D' such that (a *sigma, phi, phi) greater than or equal to 0 for every phi is an element of C-0(infinity), where *phi denotes the twisted convolution. We prove that certain boundedness for a E Y' are completely determined of the behaviour for a at origin, for example that a is an element of Y'(+), and that if a(0) < &INFIN;, then a &ISIN; L-2 &AND; L-&INFIN;. We use the results in order to determine wether positive pseudo-differential operators belong to certain Schatten-casses or not. (C) 2002 Editions scientifiques et medicales Elsevier SAS. All rights reserved.