Circles in the unit sphere of a Hilbert space

We construct on the unit sphere of a Hilbert space two closed sets F₁, F₂ such that F₁ ∩ F₂ = ? and F₁ ∩ Q ≠ ?, F₂ ∩ Q ≠ ? for any subspace Q of dimension 2.     

An infinite-dimensional statistical manifold modelled on Hilbert space

We construct an infinite-dimensional Hilbert manifold of probability measures on an abstract measurable space. The manifold, M, retains the first- and second-order features of finite-dimensional information geometry: the α-divergences admit first derivatives and mixed second derivatives, enabling the definition of the Fisher metric as a pseudo-Riemannian metric. This is enough for many applications; for example, it justifies certain projections of Markov processes onto finite-dimensional submanifolds in recursive estimation problems. M was constructed with the Fenchel–Legendre transform between Kullback–Leibler divergences, and its role in Bayesian estimation, in mind. This transform retains, on M, the symmetry of the finite-dimensional case. Many of the manifolds of finite-dimensional information geometry are shown to be $C^{\infty }$-embedded submanifolds of M. In establishing this, we provide a framework in which many of the formal results of the finite-dimensional subject can be proved with full rigour.     

The ?ojasiewicz gradient inequality in the infinite-dimensional Hilbert space framework

We provide a reasonably optimal answer to the natural question of the conditions under which an analytic function on an infinite-dimensional Hilbert space satisfies the ?ojasiewicz gradient inequality.     

Peano's theorem in an infinite-dimensional Hilbert space is false even in a weakened formulation

We formulate a continuous function FR×HH, where H is a separable Hilbert space such that the Cauchy problem. x(t)=F(t, x(t)), x(t₀)=x₀ has no solution in any neighborhood of the point t₀, no matter what t₀ R and x₀ H are considered.Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 467–477, March, 1974.In conclusion, the author thanks O. G. Smolyanov and V. I. Averbukh for their constant interest and for a number of useful remarks.     

Hilbert space representations of cross product algebras

Hilbert space representations of cross product *-algebras of the Hopf *-algebras with the coordinate algebras and of quantum vector spaces, and of with the coordinate algebras and of the corresponding quantum spheres, are investigated and classified. Invariant states on the coordinate *-algebras are described by two variants of the quantum trace.     

Homotopic invariance of infinite-dimensional homotopic groups of Hilbert space subsets

The paper discusses some equivalent ways of construction of infinite-dimensional homotopic groups of subsets and pairs of subsets in real Hilbert spaces. In the admissible class of K ₀-continuous mappings, the homotopic invariance of the mentioned groups and their isomorphism are demonstrated in the case where the basic points belong to the same component of K ₀-linear connectivity.     

Cauchy's problem for an essentially infinite-dimensional parabolic equation on an infinite-dimensional sphere

Ukrainian Mathematical Journal -     

Hilbert space representations of decoherence functionals and quantum measures

We show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.     

Optimal synthesis in an infinite-dimensional space

For a class of optimal control problems and Hamiltonian systems generated by these problems in the space l ₂, we prove the existence of extremals with a countable number of switchings on a finite time interval. The optimal synthesis that we construct in the space l ₂ forms a fiber bundle with piecewise smooth two-dimensional fibers consisting of extremals with a countable number of switchings over an infinite-dimensional basis of singular extremals.     

Hilbert space representations of general discrete time stochastic processes

We show that under mild conditions the joint densities Px₁,…,x_n) of the general discrete time stochastic process X_n on $\text{pH}$ can be computed via

$\text{Px}{}_1\text{,…,x}{}_{\mathrm{n}}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{) = 6?T(x}{}_1\text{)…T(x}{}_{\mathrm{n}}\text{)6}{}^2$

where ? is in a Hilbert space $\text{pH}$, and T (x), x ? $\text{pH}$ are linear operators on $\text{pH}$. We then show how the Central Limit Theorem can easily be derived from such representations.     

Cauchy problem for the essentially infinite-dimensional heat equation on a surface in a Hilbert space

It is proved that the Cauchy problem for a simple parabolic equation with essentially infinite-dimensional coefficients on bounded level surfaces of smooth functions in a Hilbert space is uniformly well posed.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 737–746, June, 1995.This work was supported by the Ukrainian State Committee on Science and Technology and (partially) by the International Science Foundation, Grant No. U44000.     

The size of infinite-dimensional representations

An infinite-dimensional representation π of a real reductive Lie group G can often be thought of as a function space on some manifold X. Although X is not uniquely defined by π, there are “geometric invariants” of π, first introduced by Roger Howe in the 1970s, related to the geometry of X. These invariants are easy to define but difficult to compute. I will describe some of the invariants, and recent progress toward computing them.     

The unit ball of the Hilbert space in its weak topology

We show that the unit ball of in its weak topology is a continuous image of , and we deduce some combinatorial properties of its lattice of open sets which are not shared by the balls of other equivalent norms when is uncountable.

     

Amenable representations and finite injective von Neumann algebras

Let be the unitary group of a finite, injective von Neumann algebra . We observe that any subrepresentation of a group representation into is amenable in the sense of Bekka; this yields short proofs of two known results-one by Robertson, one by Haagerup-concerning group representations into .

     

Quasiregular representations of the infinite-dimensional nilpotent group

In the present work an analog of the quasiregular representation which is well known for locally-compact groups is constructed for the nilpotent infinite-dimensional group and a criterion for its irreducibility is presented. This construction uses the infinite tensor product of arbitrary Gaussian measures in the spaces Rm with m>1 extending in a rather subtle way previous work of the second author for the infinite tensor product of one-dimensional Gaussian measures.