Latent Complete-Lattice Structure of Hilbert-Space Projectors

Abstract

To uncover the hidden complete-lattice structure of Hilbert-space projectors, which is not seen by the operator operations and relations (algebraically), resort is taken to the ranges of projectors (to subspaces—to geometry). Taking the range of a projector is completed into a bijection of all projectors onto all subspaces of any finite or countably infinite dimensional Hilbert space. As a second step, this basic bijection is upgraded into an isomorphism of partially ordered sets utilizing the sub-projector relation on the one hand, and the subspace relation on the other. As a third and final step, the basic bijection is further upgraded to isomorphism of complete lattices. The complete-lattice structure is derived for subspaces, then, using the basic bijection, it is transferred to the set of all projectors. Some consequences in the quantum-mechanical formalism are examined with particular attention to the infinite sums appearing in spectral decompositions of discrete self-adjoint operators with infinite spectra. Quanta 2019; 8: 1–10.