Quantum Stochastic Calculus on Full Fock Modules
✍ Scribed by Michael Skeide
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 351 KB
- Volume
- 173
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
We develop a quantum stochastic calculus on full Fock modules over arbitrary Hilbert B B-modules. We find a calculus of bounded operators where all quantum stochastic integrals are limits of Riemann Stieltjes sums. After having estalished existence and uniqueness of solutions of a large class of quantum stochastic differential equations, we find necessary and sufficient conditions for unitarity of a subclass of solutions. As an application we find dilations of a conservative CP-semigroup (quantum dynamical semigroup) on B with arbitrary bounded (Christensen Evans) generator. We point out that in the case B=B(G) the calculus may be interpreted as a calculus on the full Fock space tensor initial space G with arbitrary degree of freedom dilating CP-semigroups with arbitrary Lindblad generator. Finally, we show how a calculus on the boolean Fock module reduces to our calculus. As a special case this includes a calculus on the boolean Fock space.
2000 Academic Press
Contents.
1. Introduction.
- Operators on graded Banach spaces. 3. Operators on full Fock module. 4. Adaptedness. 5. Bilinear mappings on spaces of Banach space valued functions. 6. Integrals. 7. Differential equations. 8. Some 0-criteria. 9. Ito formula. 10. Unitarity conditions. 11. Cocycles. 12. Dilations. 13. The case B=B(G). 14. Boolean calculus.