An operator A ∈ B(H) whose self-commutator A∗A − AA∗ is compact is called essentially normal. Two operators A ∈ B(H) and B ∈ B(K) are said to be approximately equivalent if they are unitarily equivalent modulo compact operators; more precisely, if there is a unitary operator U : H → K such that B − UAU∗ is compact. This relation is written A ∼ B, whereas the stronger relation of unitary equival...

Assuming that quantum states, including pure states, represent subjective degrees of belief rather than objective properties of systems, the question of what other elements of the quantum formalism must also be taken as subjective is addressed. In particular, we ask this of the dynamical aspects of the formalism, such as Hamiltonians and unitary operators. Whilst some operations, such as the up...