A note on the logic of bounded quantum observables

The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators $S(H)$ on a complex Hilbert space $H$⁠, and the quantum effects for a physical system can be described by the set $E(H)$ of positive contractive operators on a complex Hilbert space $H$⁠. In this note, by the techniques of operator block and spectral, we give the simpler representation of $A∧P$ and obtained the new necessary and sufficient conditions for $A∨P$⁠, for $A∊S(H)$ and $P∊P(H)$⁠, where $P(H)$ is the set of all orthogonal projection operators on $H$⁠. In particular, we get that if $A∨P$ exists, then $A∨P∊E(H)$ for $A∊E(H)$ and $P∊P(H)$⁠. In addition, we consider the relations between the existence of $A∨B$⁠, $A−∨B−$⁠, and $A+∨B+$⁠, where $A+$⁠, $B+$⁠, $A−$⁠, and $B−$ are the positive and negative parts of $A,B∊S(H)$⁠.