Wold-type decompositions and wandering subspaces for operators close to isometries

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On Beurling-type theorems in weighted $l^2$ and Bergman spaces (2002, methodology): “More generally, it was proved in [8] that any analytic operator T satisfying the inequality (1.”
Complete Nevanlinna-Pick property of Dirichlet-type spaces (2002, background): “Indeed, in this case the operators S and Lg are Cauchy dual to each other (see [16]), therefore L=(SgS)−1 Sg and S=Lg(LLg)−1.”
A solution to the Cauchy dual subnormality problem for 2-isometries (2017, background): “(4) The notion of the Cauchy dual operator has been introduced and studied by Shimorin in the context of the wandering subspace problem for Bergman-type operators [39].”
Wold decomposition for doubly commuting isometries (2013, unknown): “[7], [8], [10], [14], [15]).”