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 q-CCR relations for |q|=1 and connections with type B-Fock spaces.
- 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.
- 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.
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, d-Gaussian processes: non-commutative 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 Meixner-type 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), 201-221.
- M. Bożejko and J. Wysoczański, Remarks on t-transformations of measures and convolutions, Ann. Inst. Henri Poincar´e-PR 37(6) (2001), 737–761.
- M. Bożejko and H. Yoshida, Generalized d-deformed Gaussian random variables, Banach Center Publ. 73 (2006), 127–140.
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