On the Space of Schwartz Operators in the Symmetric Fock Space and Its Dual

• Published in: Vestnik, St. Petersburg University. Mathematics St. Petersburg University. Mathematics, 2022-06, Vol.55 (2), p.135-140
• Main Authors: Amosov, G. G., Baitenov, E. L.
• Format: Article
• Language: English
• Subjects: Analysis; Hilbert space; Mathematical analysis; Mathematics; Mathematics and Statistics; Momentum; Operators (mathematics); Polynomials; Quantum mechanics; Quantum theory; White noise
• ISSN: 1063-4541; 1934-7855
• DOI: 10.1134/S1063454122020030

The need to work with unbounded operators is a long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider nuclear operators as the states of a quantum system and the 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) yields the average value of an observable in the fixed state of the quantum system. 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, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been accurately implemented mathematically in the Hilbert space = . The so-called space of Schwarz operators equipped with a system of seminorms and being a Frechet space was chosen as the space of states. Schwarz operators are integral operators whose kernels are functions belonging to the ordinary Schwarz space. The space dual to the space of Schwarz operators should be considered as the space of quantum observables, and it does include such standard observables as polynomials of the products of the position and momentum operators. In this work, we transfer this approach to the symmetric Fock space = over an infinite dimensional separable Hilbert space . We introduce the space of Schwartz operators in and find out which standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.