Theory of the Cauchy problem is developed for the differential-operator equations of the infinite order with totally discrete spectrum of operator. The sets of initial values of the smooth solutions are described for such equations. The correct solvability of the Cauchy problem is established on the spaces of linear functionals of the infinite order of the ultradistribution type.

[1] Gorbachuk V.I., Gorbachuk M.L. On the boundary values ​​of solutions of homogeneous differential solutions // Reports of the USSR Academy of Sciences. - 1976. - 228. N 5. - P. 1021-1024.

[2] Gorbachuk V.I., Gorbachuk M.L. Boundary values ​​of solutions of some classes of differential equations // Mat. sb. - 1977. - 102, N 1. - P. 124-150.

[3] Gorbachuk V.I., Gorbachuk M.L. Boundary value problems for differential-operator equations. - K.: Nauk. Dumka, 1984. - 283 p.

[4] Knyazyuk A. Boundary values ​​of solutions of differential equations in a Banach space // Reports of the Academy of Sciences of the Ukrainian SSR. Series A. - 1984. - N 9. - P. 12-14.

[5] Knyazyuk A. Boundary values ​​of infinitely differentiable semigroups. Preprint. - K.: Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, 1985. - N 69. - 47 p.

[6] Gorbachuk M.L., Pivtorak N.I. On solutions of evolution equations of parabolic type with degeneration // Differential equations. - 1985. - 21, N 8. - P. 1317-1324.

[7] Gorbachuk V.M., Matsishin I.T. On solutions of evolution equations with degeneration in a Banach space // Spectral theory of differential-operator equations - K.: Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, 1986. - P.5-10.

[8] Gelfand I.M., Shilov G.E. Some questions of the theory of differential equations. - M.: Fizmatgiz, 1958. - 274 p.

[9] Gorodetskyi V.V., Kolisnyk R.S. On one generalization of spaces of type $W$ // Scientific Bulletin of Chernivtsi University: 3b. scientific. pr. - Issue 134. Mathematics. - Chernivtsi: Ruta, 2002. - P.30-37.

[10] Shabat B.V. Introduction to complex analysis. - M.: Nauka, 1969. - 576 p.

[11] Gorodetsky V.V., Drin Ya.M. Parabolic pseudodifferential equations in spaces of generalized periodic functions // Supplement of the Academy of Sciences of the Ukrainian SSR. - 1991. - No. 8. - P. 18-22.