×

Commutative properties of generalized number operators. (Chinese. English summary) Zbl 1499.81068

Summary: This paper considers the commutative relations of the generalized number operator \({N_h}\) and the quantum Bernoulli noise \(\{{\partial_\sigma}, {\partial_\sigma^*}: \sigma \in \Gamma\}\) indexed by \(\Gamma\), such as Lie bracket, the expressions of the composition of \({N_h}\) and \({\partial_\sigma}( {\partial_\sigma^*})\), the commutative relation of \({N_h}\) and \({\partial_\sigma}{\partial_\sigma^*}({\partial_\sigma^*} {\partial_\sigma})\). The family of bounded linear operators \(\{{\partial_\sigma}, {\partial_\sigma^*}: \sigma \in \Gamma\}\) on \({L^2}(M)\) satisfies the canonical anti-commutative relation, nilpotency and the composition are commutative if the intersection of the index is empty. Especially, \(\{{\partial_\sigma}, {\partial_\sigma^*}: \sigma \in \Gamma\}\) satisfy “absorbing commutative relation”. In the following, the paper considers the commutative relations of \({N_h}\) and \(\{{\partial_\sigma}, {\partial_\sigma^*}: \sigma \in \Gamma\}\). For any nonnegative function \(h\) on \(\mathbb{N}\), the Lie bracket of \({N_h}\) and the \(\sigma\)-creation \({\partial_\sigma^*}\) (\(\sigma\)-annihilation \({\partial_\sigma}\)) are just \({\#_h}(\sigma){\partial_\sigma^*}({\#_h}(\sigma){\partial_\sigma})\). Especially, if the support of \(h\) is not \(\mathbb{N}\), then \({N_h}\) is commutative with some special kind of \({\partial_\sigma^*}({\partial_\sigma})\). If the support of \(h\) is a finite subset of \(\mathbb{N}\), the composition of \({N_h}\) and a special kind of \({\partial_\sigma^*}({\partial_\sigma})\) are just the creation type (annihilation type) operators. Moreover, the paper obtains that \({N_h}\) is commutative with \(\{{\partial_\sigma}{\partial_\sigma^*}, {\partial_\sigma^*} {\partial_\sigma}: \sigma \in \Gamma\}\).

MSC:

81S25 Quantum stochastic calculus
60H25 Random operators and equations (aspects of stochastic analysis)
47B44 Linear accretive operators, dissipative operators, etc.
PDFBibTeX XMLCite