The spectral theorem implies that the spectrum of a bounded normal operator acting on a Hilbert space provides a substantial information about the operator. For example, the set of eigenvalues of a normal matrix and their respective multiplicities determine the matrix up to a unitary equivalence, while the spectral measure, \(E_B(\lambda )\) of a normal operator B acting on a Hilbert space determines B via the integral spectral resolution, \(\begin{aligned} B=\int _{\sigma (B)} \lambda dE_B(\lambda ). \end{aligned}\) In general, for a non-normal operator the spectrum provides a rather limited information about the operator. In this paper we show that, if we include an arbitrary bounded operator B acting on a separable Hilbert space into a quadruple which contains 3 specific operators along with B, it is possible to reconstruct B from the proper projective joint spectrum of the quadruple (and here we mean reconstruct precisely, not up to an equivalence). We call this process spectral reconstruction.
