On the invertibility of infinite-dimensional pseudodifferential operators (Q2487452)

The authors obtain some conditions of one-valued solvability of some infinite-dimensional pseudodifferential equations and construct some classes of invertible pseudodifferential operators with symbols depending on two arguments \((n,y)\in H\times H\), where \(H\) is an infinite-dimensional Hilbert space. Such operators define continuous mappings in the space of test functions \(Z\), and their adjoint operators acting in the space of generalized measures \(Z'\) are infinite-dimensional PDOs coinciding with standard PDOs in the finite-dimensional case. The main tools of the proof are the method of successive approximations and the Fourier transform.