Marek Bożejko
Generalized Fock spaces and second quantization as examples of
hypercontractive semigroups
In my talk we will consider the following topics:
 Generalized Fock spaces F(T,H) for real contraction T on a real Hilbert space H.
 Anyonic Fock spaces and qCCR relations for q=1 and connections with type BFock spaces.
 Second quantization Γ(T,S) from the von Neumann algebra G(T,H) generalized by TGaussian operators G(f) = a(f) + a^{+}(f), where f is in a real Hilbert space H.
 If S = exp(t) Id, we get so called generalized OrnsteinUhlenbeck semigroup
U_{t} = Γ(T,S) and in many cases of T we have ultrarcontactivity, i.e.
U_{t} maps L^{2} into L^{∞} , where here L^{2} = 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.
References:
 M. Bożejko and M. Guta, Functors of white noise associated to characters of the infinite symmetric group, Comm. Math. Phys. 229 (2002), 209–227.
 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.
 M. Bożejko, B. K¨ummerer and R. Speicher, dGaussian processes: noncommutative and classical aspects, Comm. Math. Phys., 185(1), (1997), 129–154.
 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.
 M. Bożejko, E.Lytvynov, I.Rodionova, An extended Fock space and noncommutative Meixnertype orthogonal polynomials, Uspiekhi Mat.Nauk,70(5),2015,47 pp. ,there is also russian version.
 M. Bozejko and R. Speicher, An example of a generalized Brownian motion, Commun. Math. Phys. 137 (1991), 519–531.
 M. Bożejko and R. Speicher, Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces, Math. Ann. 300 (1994), 97–120.
 M. Bożejko and R. Speicher, Interpolations between bosonic and fermionic relations given by generalized Brownian motions, Math. Z. 22 (1996), 135–159.
 M. Bożejko and R. Szwarc, Asymptotic Combinatorics with Applications to Mathematical Physics, Lecture Notes in Mathematics 1815, (2003), 201221.
 M. Bożejko and J. Wysoczański, Remarks on ttransformations of measures and convolutions, Ann. Inst. Henri Poincar´ePR 37(6) (2001), 737–761.
 M. Bożejko and H. Yoshida, Generalized ddeformed Gaussian random variables, Banach Center Publ. 73 (2006), 127–140.
