Continuity and compactness properties of pseudo-differential operators

Consider a Hormander' symbol class S(mu, g) such that mu h(n) is bounded for a suitable n >= 0. Then we show that the Weyl quantization of a is an element of S(mu, g) is bounded in L-2 if and only if the symbol a is bounded. If mu h(n) -> 0 as |x| + |xi| -> infinity, then we show that a(w) is compact in L-2 if and only if a -> 0 as |x| + |xi| -> infinity.