An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

作者：Bozejko M^*; Lytvynov E W; Rodionova I V

来源：Russian Mathematical Surveys, 2015, 70(5): 857-899.

DOI：10.1070/RM2015v070n05ABEH004965

• anyon commutation relations
• anyon Fock space
• gamma process
• Jacobi field
• Levy white noise
• Meixner class of orthogonal polynomials

Let nu be a finite measure on R whose Laplace transform is analytic in a neighbourhood of zero. An anyon Levy white noise on (R-d, dx) is a certain family of noncommuting operators a omega, phi o on the anyon Fock space over L-2 (R-d x R, dx A nu), where phi = phi(x) runs over a space of test functions on R-d, while omega = omega(x) is interpreted as an operator-valued distribution on R-d. Let L-2 (tau) be the noncommutative L-2-space generated by the algebra of polynomials in the variables a omega, phi O, where tau is the vacuum expectation state. Noncommutative orthogonal polynomials in L-2 (tau) of the form aP(n)(omega), f((n))o are constructed, where f((n)) is a test function on (R-d)(n), and are then used to derive a unitary isomorphism U between L-2 (tau) and an extended anyon Fock space F(L-2 (R-d, dx)) over L-2 (R-d, dx). The usual anyon Fock space F(L-2 (R-d, dx)) over L-2 (R-d, dx) is a subspace of F(L-2 (R-d, dx)). Furthermore, the equality F(L-2 (R-d, dx)) - F(L-2 (R-d, dx)) holds if and only if the measure nu is concentrated at a single point, that is, in the Gaussian or Poisson case. With use of the unitary isomorphism U, the operators a omega, phi o are realized as a Jacobi (that is, tridiagonal) field in F(L-2 (R-d, dx)). A Meixner-type class of anyon Levy white noise is derived for which the corresponding Jacobi field in F(L-2 (R-d, dx)) has a relatively simple structure. Each anyon Levy white noise of Meixner type is characterized by two parameters, lambda is an element of R and eta >= 0. In conclusion

• 出版日期2015