Symmetric Hilbert spaces arising from species of structures |
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Authors: | M?d?lin Gu?a Hans Maassen |
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Institution: | (1) Mathematisch Instituut, Katholieke Universiteit Nijmegen, Toernooiveld 1, 6526 ED Nijmegen, The Netherlands (e-mail: guta@sci.kun.nl, maassen@sci.kun.nl) , NL |
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Abstract: | Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some lsquo;one particle space’ are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization
by building on Joyal's notion of a combinatorial species. Any such species F gives rise to an endofunctor of the category of Hilbert spaces with contractions mapping a Hilbert space to a symmetric Hilbert space with the same symmetry as the species F. A general framework for annihilation and creation operators on these spaces is developed, and compared to the generalised
Brownian motions of R. Speicher and M. Bożejko. As a corollary we find that the commutation relation with admits a realization on a symmetric Hilbert space whenever f has a power series with infinite radius of convergence and positive coefficients.
Received: 7 April 2000; in final form: 28 November 2000 / Published online: 19 October 2001 |
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