On compactness of the difference of composition operators

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Linear relations in the Calkin algebra for composition operators (2006, background): “Most recently, Nieminen and Saksman [19] have shown that the justquoted condition of J. E. Shapiro on singular parts, plus uniform integrability of the differences of the absolutely continuous densities of the respective Clark measures of ϕ and ψ, are together equivalent to compactness of Cϕ − Cψ on”
Compact Differences of Composition Operators on Bloch and Lipschitz Spaces (2007, background): “Most papers in this area have focused on (weighted) Bergman and Dirichlet spaces and especially the Hardy space H2; see e.g. [10], [23], [21], [17], [9] and [ 18 ].”
On the connected component of compact composition operators on the Hardy space (2007, methodology): “In particular, a characterization for compact differences of composition operators on M and L(1) was found in [19].”
Compact Differences of Composition Operators on the Bergman Spaces Over the Ball (2013, background): “We remark that the Hardy space analogue for the compact difference has been also known over the disk, but with the restriction 1 ≤ p < ∞; see [13].”