## Quantum stochastic operator cocycles via associated semigroups.(English)Zbl 1122.60061

This paper treats the functional equation $$V_{r+t} = V_r \sigma_r (V_t)$$ for all $$r, t \geq 0$$ with $$V_0 = I$$, for a family of contractions on $${\mathfrak h} \otimes {\mathcal F}$$ adapted to the Fock operator filtration. Here $${\mathcal F}$$ is the symmetric Fock space over $$L^2({\mathbb R}_+ ; \mathfrak k)$$, $${\mathfrak h}$$ and $${\mathfrak k}$$ are fixed arbitrary Hilbert spaces, and $$( \sigma_t )_{ t \geq 0}$$ is the endomorphism semigroup of shifts, amplified to $$B( {\mathfrak h} \otimes {\mathcal F})$$. Such a family is called a left contraction cocycle on $${\mathfrak h}$$ with noise dimension space $${\mathfrak k}$$. Contraction cocycles can be constructed by solving quantum stochastic differential equations (QSDEs) of Hudson-Parthasarathy type. There is a characterization (Theorem 1.6, p. 544) of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated family of semigroups, namely,
(a) “there is a left contraction cocycle $$V$$ on $${\mathfrak h}$$ whose associated family of semigroups includes $${\mathcal Q}_{\text{T}}=\{ Q^{c,d} ; c,d \in \text{{T}} \}$$”
is equivalent to
(b) “for all $$n \in {\mathbb N}$$, $$Y \in M_n( | {\mathfrak h} \rangle)=B( {\mathbb C}^n ; {\mathfrak h}^n )$$, and positive invertible matrices $$A, B \in$$ $$M_n({\mathbb C})$$, if $$\| A^{- 1/2} Y B^{ -1/2} \|$$ $$\leqslant 1$$, then
$\| ( A \bullet \bar{\omega}_t^{\mathbf{c}} )^{- 1/2} ( Q_t^{\mathbf{c}} \bullet Y) ( B \bullet \bar{\omega}_t^{\mathbf{c}} ) ^{-1/2} \| \leqslant 1$
holds for all $$\mathbf c\in \text{{T}}^n, t \geq 0$$”, where
$${\mathcal Q}_{\text{{T}}}$$ is a family of semigroups on $${\mathfrak h}$$ indexed by a total subset {T} of $${\mathfrak k}$$ which contains 0, $$| {\mathfrak h} \rangle:=B({\mathbb C} ; {\mathfrak h})$$ is the column operator space determined by $${\mathfrak h}$$ [cf. G. Pisier, “Introduction to operator space theroy” (2003; Zbl 1093.46001)], $$\overline\omega_t^{c,d}=\exp - t\chi (c,d)$$ for given $$c,d \in{\mathfrak k}$$, $$\overline\omega_t^{\mathbf c}:= [\overline\omega_t^{c_i, c_j} ]\in M_n({\mathbb C})=B( {\mathbb C}^n)$$ for given $${\mathbf c}\in{\mathfrak k}^n$$, $$Q_t^{\mathbf c}:=[ Q_t^{c_i, c_j} ]\in M_n( B({\mathfrak h}))=B({\mathfrak h}^n )$$, and the symbol $$\bullet$$ denotes the Schur product of matrices.
By means of this remarkable characterization of such cocycles through semigroups, the authors provide a new method of constructing cocycles which in turn leads to new existence results for QSDEs, because the above characterization yields indeed a general principle for the construction of such cocycles by approximation of their stochastic generators. When the driving noise is infinite-dimensional, then the coefficient of a QSDE is naturally given as a sesquilinear operator-valued map or, in terms of a coordinate system for the noise dimension space $${\mathfrak k}$$, as an infinite matrix $$[ F_{\beta}^{\alpha}]$$. Moreover, the authors show that, if a process satisfying such a form QSDE is contractive and strongly measurable, then the coefficient is necessarily given by an operator, equivalently the matrix must be semiregular. Lastly, necessary and sufficient conditions, of weak differentiability type, for a strongly continuous left contraction cocycle to satisfy such a QSDE, are given as well.
This paper is basically due to L. Accardi (ed.) and W. von Waldenfels (ed.) [“Quantum probability and applications IV” (1989; Zbl 0672.00013)], A. Mohari [Sankhyā, Ser. A 53, No. 3, 255–287 (1991; Zbl 0751.60062)], and F. Fangnola [“Quantum probability and related topics VIII (ed. L. Accardi), 143–164 (1993)], extending known results for Markov-regular cocycles and QSDEs with bounded coefficients by the authors’ previous works [Probab. Theory Relat. Fields 116, No. 4, 505–543 (2000; Zbl 1079.81542) and J. Funct. Anal. 178, No. 2, 269–305 (2000; Zbl 0969.60066)].
Some remarkable features of the present paper consist of the followings: (i) their development of the theory is coordinate-free; (ii) no separability assumptions are imposed on either the initial space $${\mathfrak h}$$ or the noise dimension space $${\mathfrak k}$$. For other related works, see, e.g., J. M. Lindsay [in: Quantum independent increment processes. I. From classical probability to quantum stochastic calculus. Lect. Notes Math. 1865, 181–271 (2005; Zbl 1072.81039)], and the authors’ recent work [Proc. Indian Acad. Sci., Math. Sci. 116, No. 4, 519–529 (2006; Zbl 1123.47052)]. A different approach to the characterization and construction of cocycles through semigroup methods can be found in [V. Liebscher, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4, No. 2, 215–219 (2001; Zbl 1057.47509)].

### MSC:

 60H25 Random operators and equations (aspects of stochastic analysis) 47B80 Random linear operators 81S25 Quantum stochastic calculus 47D06 One-parameter semigroups and linear evolution equations 34G10 Linear differential equations in abstract spaces
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