We prove a general spectral theorem for an arbitrary densely defined closed linear operator $T$ between complex Hilbert spaces $H$ and $K$. The corresponding operator measure is partial isometry valued and has properties similar to those of the resolution of the identity of a non-negative self-adjoint operator. The main method is the use of the canonical factorization (polar decomposition) obtained by v. Neumann and Murray. The uniqueness of the generalized resolution of the identity is studied together with the properties of a (non-multiplicative) functional calculus. The properties of this generalized resolution of the identity are also investigated.
"Partial isometries and a general spectral theorem." Adv. Oper. Theory 4 (2) 351 - 368, Spring 2019. https://doi.org/10.15352/aot.1804-1355