Rights statement: The final, definitive version of this article has been published in the Journal, Journal of Mathematical Analysis and Applications 409 (2), 2013, © ELSEVIER.
Accepted author manuscript, 481 KB, PDF document
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Sesquilinear quantum stochastic analysis in Banach space. / Lindsay, Martin; Das, Bata Krishna; Tripak, Orawan.
In: Journal of Mathematical Analysis and Applications, Vol. 409, No. 2, 15.01.2014, p. 1031-1051.Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - Sesquilinear quantum stochastic analysis in Banach space
AU - Lindsay, Martin
AU - Das, Bata Krishna
AU - Tripak, Orawan
N1 - The final, definitive version of this article has been published in the Journal, Journal of Mathematical Analysis and Applications 409 (2), 2013, © ELSEVIER.
PY - 2014/1/15
Y1 - 2014/1/15
N2 - A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.
AB - A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.
KW - Quantum stochastic differential equation
KW - Quantum Wiener integral
KW - Quantum stochastic cocycle
KW - Trotter product formula
U2 - 10.1016/j.jmaa.2013.01.067
DO - 10.1016/j.jmaa.2013.01.067
M3 - Journal article
VL - 409
SP - 1031
EP - 1051
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 2
ER -