Operators and POVMs generated by Parseval frames

Abstract Let $${\mathcal {F}}_e$$ F e be a Parseval frame in a Hilbert space $${\mathcal {H}}$$ H and let $$\textsf{E}$$ E be a set of real numbers. From these data, we construct an operator $$H_{\textsf{E},e}$$ H E , e and a positive operator-valued measure (POVM) $$F_{\textsf{E},e}$$ F E , e This paper investigates in detail the relationship between the operator $$H_{\textsf{E},e}$$ H E , e and the POVM $$F_{\textsf{E},e}$$ F E , e . Our results extend the classical correspondence between a self-adjoint operator generated by an orthonormal basis and its associated projection-valued (spectral) measure.