XXXV Workshop on Geometric Methods in Physics 26.06-2.07.2016

Marek Bożejko

Generalized Fock spaces and second quantization as examples of hypercontractive semigroups

In my talk we will consider the following topics:
  1. Generalized Fock spaces F(T,H) for real contraction T on a real Hilbert space H.
  2. Anyonic Fock spaces and q-CCR relations for |q|=1 and connections with type B-Fock spaces.
  3. Second quantization Γ(T,S) from the von Neumann algebra G(T,H) generalized by T-Gaussian operators G(f) = a(f) + a+(f), where f is in a real Hilbert space H.
  4. If S = exp(-t) Id, we get so called generalized Ornstein-Uhlenbeck semigroup Ut = Γ(T,S) and in many cases of T we have ultrarcontactivity, i.e. Ut maps L2 into L , where here L2 = the Fock space F(T,H) and L = G(T,H).

The subject of the talk is taken from the main results of the papers together with W. Ejsmont, T. Hasebe, E. Lytvynov, I. Rodionova, Q. Xu and J. Wysoczański.

  1. M. Bożejko and M. Guta, Functors of white noise associated to characters of the infinite symmetric group, Comm. Math. Phys. 229 (2002), 209–227.
  2. M. Bożejko, W. Ejsmont and T. Hasebe, Fock space associated to Coxeter groups of type B, Journal of Functional Analysis J. Funct. Anal. 269(6) (2015), 1769–1795.
  3. M. Bożejko, B. K¨ummerer and R. Speicher, d-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys., 185(1), (1997), 129–154.
  4. M. Bożejko, E. Lytvynov and J. Wysoczański, Noncommutative L´evy processes for generalized (particularly anyon) statistics, Comm. Math. Phys. 313, Issue 2 (2012), 535–569.
  5. M. Bożejko, E.Lytvynov, I.Rodionova, An extended Fock space and noncommutative Meixner-type orthogonal polynomials, Uspiekhi Mat.Nauk,70(5),2015,47 pp. ,there is also russian version.
  6. M. Bozejko and R. Speicher, An example of a generalized Brownian motion, Commun. Math. Phys. 137 (1991), 519–531.
  7. M. Bożejko and R. Speicher, Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces, Math. Ann. 300 (1994), 97–120.
  8. M. Bożejko and R. Speicher, Interpolations between bosonic and fermionic relations given by generalized Brownian motions, Math. Z. 22 (1996), 135–159.
  9. M. Bożejko and R. Szwarc, Asymptotic Combinatorics with Applications to Mathematical Physics, Lecture Notes in Mathematics 1815, (2003), 201-221.
  10. M. Bożejko and J. Wysoczański, Remarks on t-transformations of measures and convolutions, Ann. Inst. Henri Poincar´e-PR 37(6) (2001), 737–761.
  11. M. Bożejko and H. Yoshida, Generalized d-deformed Gaussian random variables, Banach Center Publ. 73 (2006), 127–140.

Event sponsored by:
National Science Foundation          Belgian Science Policy Office          University of Bialystok

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