Maximal vectors in Hilbert space and quantum entanglement

Let V be a norm-closed subset of the unit sphere of a Hilbert space H that is stable under multiplication by scalars of absolute value 1. A maximal vector (for V) is a unit vector ξ∈H whose distance to V is maximum

     

On the Hamiltonian property of linear dynamical systems in Hilbert space

Mathematical Notes - Conditions for the operator differential equation $$dot x = Ax$$ possessing a quadratic first integral (1/2)(Bx, x) to be Hamiltonian are obtained. In the finite-dimensional...     

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.     

Dimensional renormalization inp-adic models of field theory

We define fractional-dimensional p-adic Feynman amplitudes and construct a dimensional renormalization with minimum subtractions. In the fermionic model case, another dimensional renormalization procedure is defined as the inversion of the normalizing transformation at the trivial stable point for the hierarchical renormalization group transformation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 462–475, June, 2000.     

Functional equations and renormalizations inp-adic field theories

The procedure of constructing the discrete approximation of p-adic ⁴-theory on a hierarchical lattice is reduced to the solution of an integral functional equation. The renormalization procedure and -functions are constructed in terms of this solution.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No 1, pp. 3–16, October, 1996.     

Sampling in a Hilbert space

An analog of the Whittaker-Shannon-Kotel'nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.

     

Locating subsets of a Hilbert space

This paper deals with locatedness of convex subsets in inner product and Hilbert spaces which plays a crucial role in the constructive validity of many important theorems of analysis.

     

Limit-ill-posed equations in a Hilbert space

The behavior of the solution of a limit-ill-posed problem on fixed compacta is investigated for integral operators, acting in a Hilbert space.Translated from Ukrainskii Matematicheskii Zhurnal, vol. 43, No. 2, pp. 241–247, February, 1991.     

Projection lattices in a Hilbert space

Vologda Polytechnics Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 2, pp. 86–87, April–June, 1989.     

Closure in a Hilbert space of a prehilbert space Chebyshev set

Let E denote the real inner product space that is the union of all finite dimensional Euclidean spaces. There is a bounded nonconvex set S, that is a subset of E, such that each point of E has a unique nearest point in S. Let H denote the separable Hilbert space that is the completion of space E. A condition is given in order that a point in H have a unique nearest point in the closure of S. We shall also provide an example where the condition fails.     

Rearrangements of series in a Hilbert space

A theorem on rearrangements of numerical series, proved by Agnew, is extended to series in a Hilbert space. A complete proof is given of Orlicz's theorem on unconditionally convergent series in a Hilbert space.Translated from Matematicheskie Zametki, Vol. 12, No. 3, pp. 275–280, September, 1972.     

Hyperbolic Hilbert space

The hyperbolic complex space is one class of non-Euclidean spaces with continuous singular points. It corresponds with Minkowski space, and it has the characteristic that the space-time direction is different in nature. Regard the hyperbolic complex space as original spaces. We can abstract a class of the hyperbolic inner product space and the hyperbolic Hilbert space.