Modifying a resolution of the identity by the action of a linear operator

The main purpose of this paper is to investigate the behavior of the Hilbert space valued functions [Formula: see text] that are obtained by applying a linear operator [Formula: see text] to a resolution of the identity (i.e. a continuous Parseval frame) [Formula: see text]. After reviewing the case when [Formula: see text] is bounded with bounded inverse, we study, in more detail, generalized Riesz maps, i.e. the image of a [Formula: see text] by an invertible closed linear operator [Formula: see text].