On P,-Boundedness and Compactness V of Linear Pseudo. Differential Operators -V J. TERVO . 0 V V V Boundedness and compactness arguments in the Hörmander spaces a p.k for linear pseudodifferential operators L(X, D)'are considered. The symbol L(x, ) of L(X, D) is assumed to obey appropriate temperate criteria, which guarantee that L(X, D) maps the Schwartz class X into itself and,that the formal transpose L'(X, D): Y -* a" exists. A characterization V for the boundedness of the operator L'(X,

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... ; -is obtained. Asufficient condition for the boundedness of the operatorL'(X, D): 'pkk -',.k with p E (1, oo) is established as well. Finally, the compactness of the continuous extension of L'(X, D): Vpkk-(G) \ 'p.k is studied, where GO an open bounded set in R' and where '-p.kk-(G) is (essentially)