Infinite-dimensional Separable Space Research Articles

On the space of Schwartz operators in the symmetric Fock space and its dual

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is the need to work with unbounded operators. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider the states of a quantum system as nuclear operators, and consider bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) gives the average value of a quantum observable in a fixed state. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum operators, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been mathematically correct implemented in the Hilbert space H = L^2(R^N). The so-called space of Schwarz operators, equipped with a system of semi-norms and being a Frechet space, was chosen as the space of states. Schwarz operators appeared to be integral operators whose kernels are functions belonging to the usual Schwarz space. The dual space to the space of Schwarz operators should be considered as the space of quantum observables and it really includes such standard observables as polynomials from the products of the position and momentum operators. In the present work we transfer this approach to the symmetric Fock space H = F(H) over an infinite dimensional separable Hilbert space H. We introduce the space of Schwartz operators in F(H) over an infinite dimensional separable space H and investigate which of the standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.

Noncircular elliptic discs as numerical ranges of nilpotent operators

We show that (1) if A is a nonzero quasinilpotent operator with ran A n closed for some n ≥ 1, then its numerical range W(A) contains 0 in its interior and has a differentiable boundary, and (2) a noncircular elliptic disc can be the numerical range of a nilpotent operator with nilpotency 3 on an infinite-dimensional separable space. (1) is a generalization of the known result for nonzero nilpotent operators, and (2) is in contrast to the finite-dimensional case, where the only elliptic discs which are the numerical ranges of nilpotent finite matrices are the circular ones centred at the origin.

Band-diagonal operators

We show that there are operators on a five-dimensional Hilbert space which are not tridiagonal, and that there are compact operators and normal operators on separable infinite-dimensional spaces which are not band-diagonal.