Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis

Pseudodifferential equations of the formv(D_x)y=f (here,v is a function holomorphic at zero andD _x is a pseudodifferential operator) are studied on spaces of test functions of non-Gaussian infinite-dimensional analysis. The results obtained are applied to the construction of a generalized translation operatorT _y ^χ =χ(<y, D_χ>) on the spaces indicated and to the investigation of its properties. In particular, we prove the associativity, commutativity, and other properties ofT _y ^χ , which are analogs of the classical properties of a generalized translation operator. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1334–1341, October, 1999.