A flow on a C*-algebra $A$ is a group homomorphism $\sigma $ from ${\mathbb R}$ into the group of *-automorphisms of $A$ such that the map $$ t\in {\mathbb R}\mapsto \sigma _t(a)\in A $$ is norm-continuous for every $a$ in $A$. If $\rho $ is another flow on $A$, one says that $\sigma $ and $\rho $ are cocycle-conjugate if there exists a norm-continuous function $u$ from ${\mathbb R}$ to the group of unitary elements of $M(A)$, the multiplier algebra of $A$, such that $$ \rho _t(a) = u_t\sigma _t(a)u_t^*, $$ for all $t$ in ${\mathbb R}$, and all $a$ in $A$, and moreover such that $$ u_{t+s} = u_t\sigma _t(u_s), $$ for all $t$ and $s$ in ${\mathbb R}$.
If $K(H)$ denotes the algebra of all compact operators on Hilbert's space $H$, a consequence of Stone's Theorem states that for every flow $\sigma $ on $K(H)$ there exists a (possibly unbounded) self-adjoint operator $h$ on $H$ such that $\sigma =\sigma ^h$, where $$ \sigma ^h_t(a) = e^{ith}ae^{-ith}, $$ for all $t$ in ${\mathbb R}$, and $a$ in $K(H)$.
Given two self-adjoint operators $h$ and $k$, a simple sufficient condition for $\sigma ^h$ and $\sigma ^k$ to be cocycle-conjugate on $K(H)$ is that $h$ and $k$ be unitarily conjugate. Another is that $h$ and $k$ commute (in the sense that their spectral projections commute) and the difference $h-k$ is bounded.
My question regards the classification of flows on $K(H)$ up to cocycle-conjugacy.
Question. Given two self-adjoint operators $h$ and $k$, find a sensible necessary and sufficient condition in terms of $h$ and $k$ in order for $\sigma ^h$ and $\sigma ^k$ to be cocycle-conjugate.
If this proves to be difficult, any further sufficient conditions for cocycle-conjugacy would also be appreciated.
