Some basic problems related to closed graph theorems and, in general, to homological properties of locally convex spaces, which attracted the experts at the beginning of 70-th, were solved. Those problems tracing back to Dieudonne, L. Schwartz, Grothendieck, Koethe, Ptak, Kelley, Raikov were already 15–20 years old that time. In particular it has been constructed a quotient of the space $D(R)$ which is isomorphic to a proper dense subspace of $R^\infty$ (and hence non-complete and metrizable). It has been also shown that the Pontryagin duality between the spaces $D(R)$ and $D'(R)$ can not be extended to their subspaces and quotients. In solving all of those problems a method of effective constructing, in locally convex spaces, some sequentially closed non-closed subsets of different types (countable, convex, vector subspaces etc.) has been derived. This method has also an independent interest and has been used to solve some other problems, for example to construct infinitely differentiable discontinuous functions on locally convex spaces (M. O. Smolyanova). Besides by this method it has been possible to solve, using some properties of spaces $D(R)$ and $D'(R)$, five of twelve problems posed in the famous paper of Dieudonne and L. Schwartz. By that time those problems have been already solved by Grothendieck who needed, without this method, to construct for that purpose some special locally convex spaces. It has been proved (together with A. V. Uglanov) that the Wiener measure does not have any Hilbert support; this statement refutes a conjecture of F. A. Berezin according to which the $\sigma$-additivity of the Wiener measure can be deduced from the Minlos–Sazonov theorem. It has been shown (together with E. T. Shavgulidze) that the Hamiltonian Feynman measure (on the set of paths in the phase space) can be considered as an analytical continuation of a Gaussian measure; this statement refutes another Berezin's conjecture. Some infinite dimensional pseudodifferential operators with $pq$- and $qp$-symbols have been defined and (together with A. Yu. Khrennikov) an algebra of such operators have been constructed; by this way one more Berezin's problem have been solved. A theory of smooth functions and (together with S. V. Fomin) a theory of smooth measures on infinite dimensional spaces have been developed. It has been shown (together with J. Kupsch) that there does not exist any Hilbert norm on a tensor algebra (including the Grassman algebra) that satisfies the estimate $\|xy\|\le c\|x\|\|y\|$ with the constant $c=1$ but such a norm has been constructed for $c=\sqrt{3}$; this means that a problem tracing back to B. deWitt is solved. Some representations of solutions for stochastic Schroedinger–Belavkin equations by Feynman path integrals are obtained (together with S. Albeverio, V. M. Kolokol'tsov, A. Truman). It is proved (together with M. O. Smolyanova) a Prigogine's conjecture about irreducibility of Liouvillian dynamics to Hamiltonian dynamics. Some connections between Levy Laplacians and (quantum) stochastic processes are described (together with L. Accardi). Surface measures on trajectories in Riemannian submanifolds of Euclidian spaces (and Riemannian manifolds) generated by measures on trajectories in enveloping spaces are introduced and (together with H. v. Weizsaecker) their properties are investigated. In particular it is shown that in the case of the Wiener measure on trajectories in the enveloping manifold the corresponding surface measure is equivalent to the Wiener measure on trajectories in the submanifolds and the corresponding density is calculated. Some Feynman and Feynman–Kac formulas for solutions of Schroedinger equations on Riemannian manifolds (including stochastic equations) are obtained (together with A. Truman); by this way some problems tracing back to C. deWitt-Morett and D. Elworthy are solved.
Main publications:
Kupsch J., Smolyanov O. G. Functional representations for Fock superalgebras // Infinite Dimensional Analysis, Quantum Probability and Related Topics, v. 1, no. 2, 1998, 285–324.
Smolyanov O. G., Weizsaecker H. v., Wittich O. Brownian motion on a manifiold as limit of stepwise conditioned standard Brownian motions // Canadian Mathematical Society Conference Proceedings, v. 29, 2000, 589–602.
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