Structured Parseval Frames in Hilbert $C^*$-modules

摘要

We investigate the structured frames for Hilbert $C^{*}$-modules. In the casethat the underlying $C^{*}$-algebra is a commutative $W^*$-algebra, we provethat the set of the Parseval frame generators for a unitary operator group canbe parameterized by the set of all the unitary operators in the doublecommutant of the group. Similar result holds for the set of all the generalframe generators where the unitary operators are replaced by invertible andadjointable operators. Consequently, the set of all the Parseval framegenerators is path-connected. We also obtain the existence and uniquenessresults for the best Parseval multi-frame approximations for multi-framegenerators of unitary operator groups on Hilbert $C^*$-modules when theunderlying $C^{*}$-algebra is commutative.