We study general continuous properties for an increasing family of Banach spaces of classes for pseudo-differential symbols, where the largest symbol space was introduced by J. Sjöstrand 1993. We prove that corresponding pseudo-differential operators are contained in the some certain sets of Schatten-von Neumann operators. We prove also that one obtains Hölder relations from the operator product and the usual multiplication, and that the convolution multiplication give rise to some Young type relations. Some further extensions are also discussed