A note on the infimum problem of Hilbert space effects

The quantum effects for a physical system can be described by the set E ( H ) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I . The infimum problem of Hilbert space effects is to find under what condition the infimum A ∧ B exists for two quantum effects A and B ∊ E ( H ) . The problem has been studied in different contexts by Kadison, Gudder, Moreland, and Ando. In this Note, using the method of the spectral theory of operators, we give a affirmative answer of a conjecture of [S. Gudder, J. Math. Phys. 37, 2637–2642 (1996)]. In addition, some properties of generalized infimum A ⨅ B were considered.