Self-adjoint operators in the Smolyanov–Shamarov space

We prove the essential self-adjointness of the Laplace–Volterra operator in the Smolyanov–Shamarov space, i.e., in the space of functions defined on a real infinite-dimensional separable Hilbert space and square-integrable with respect to a generalized Lebesgue–Feynman–Smolyanov–Shamarov measure. To prove this, we first prove the essential self-adjointness of the operator of multiplication by a quadratic function with a kernel operator. Then we apply the infinite-dimensional Fourier transform, mapping functions from an infinite-dimensional analogue of the Schwartz space to functions from the same space. Furthermore, a consequence of one of the proved theorems is the separability of the Smolyanov–Shamarov space.