Harmonic functions on quadrature domains and Denjoy-type domains

Joanna Pres
2010 Irish Mathematical Society Bulletin  
This is an abstract of the PhD thesis Supersymmetric Quantum Stochastic Analysis written by Clodagh Carroll under the supervision of Dr. Stephen Wills at the School of Mathematical Sciences, UCC and submitted in June 2010. Z 2 -graded quantum stochastic calculus is reformulated into a basis independent, infinite-dimensional calculus, expanding on the finite dimensional calculus developed by T.M.W. Eyre and R.L. Hudson. We follow J.M. Lindsay's synthesis, in the sense of Hudson and
more » ... making use of the Hitsuda-Skorohod integral and Operator Space Theory to develop this calculus. Z 2 -graded integrals are expressed in terms of Bosonic integrals and the fundamental formulae and estimate of Z 2 -graded quantum stochastic calculus are derived, followed by their higher order analogues. Two types of quantum stochastic differential equation are analysed; the first is the Hudson-Parthasarathy equation with bounded generators F t , L and bounded operators Ψ, Φ. Existence and uniqueness results for (1) and (2) are proved, with suitable local uniform boundedness and regularity conditions imposed. Necessary and sufficient conditions for isometric, co-isometric and unitary solutions are derived for each differential equation and the relationship between solutions of (1) and (2) is examined. 24 Abstracts of PhD Theses Existence and uniqueness of solutions of the quantum stochastic differential equation
doi:10.33232/bims.0066.32.34 fatcat:vkw64ppzkze4dn4fp456pfssdi