8 research outputs found
Generalized Fock Spaces and the Stirling Numbers
The Bargmann-Fock-Segal space plays an important role in mathematical physics and has been extended into a number of directions. In the present paper, we imbed this space into a Gelfand triple. The spaces forming the Fréchet part (i.e., the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and Salomon
Realizations of non-commutative rational functions around a matrix centre, II: The lost-abbey conditions
In a previous paper the authors generalized classical results of minimal
realizations of non-commutative (nc) rational functions, using nc
Fornasini--Marchesini realizations which are centred at an arbitrary matrix
point. In particular, it was proved that the domain of regularity of a nc
rational function is contained in the invertibility set of a corresponding
pencil of any minimal realization of the function. In this paper we prove an
equality between the domain of a nc rational function and the domain of any of
its minimal realizations. As for evaluations over stably finite algebras, we
show that the domain of the realization w.r.t any such algebra coincides with
the so called matrix domain of the function w.r.t the algebra. As a corollary
we show that the domain of regularity and the stable extended domain coincide.
In contrary to both the classical case and the scalar case -- where every
matrix coefficients which satisfy the controllability and observability
conditions can appear in a minimal realization of a nc rational function -- the
matrix coefficients in our case have to satisfy certain equations, called
linearized lost-abbey conditions, which are related to Taylor--Taylor
expansions in nc function theory
White Noise Space Analysis and Multiplicative Change of Measures
In this paper, we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogs of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations
White Noise Space Analysis and Multiplicative Change of Measures
In this paper we display a family of Gaussian processes, with explicit
formulas and transforms. This is presented with the use of duality tools in
such a way that the corresponding path-space measures are mutually singular. We
make use of a corresponding family of representations of the canonical
commutation relations (CCR) in an infinite number of degrees of freedom.
A key feature of our construction is explicit formulas for associated
transforms; these are infinite-dimensional analogues of Fourier transforms. Our
framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert
spaces, and Fock spaces. The latter forms the setting for our CCR
representations. We further show, with the use of representation theory, and
infinite-dimensional analysis, that our pairwise inequivalent probability
spaces (for the Gaussian processes) correspond in an explicit manner to
pairwise disjoint CCR representations
Generalized Fock spaces and the Stirling numbers
The Bargmann-Fock-Segal space plays an important role in mathematical
physics, and has been extended into a number of directions. In the present
paper we imbed this space into a Gelfand triple. The spaces forming the
Fr\'echet part (i.e. the space of test functions) of the triple are
characterized both in a geometric way and in terms of the adjoint of
multiplication by the complex variable, using the Stirling numbers of the
second kind. The dual of the space of test functions has a topological algebra
structure, of the kind introduced and studied by the first named author and G.
Salomon.Comment: revised versio