Multiplication operators on the Bergman space via the Hardy space of the bidisk

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Reducing subspaces for analytic multipliers of the Bergman space (2011, methodology): “[8, 11], which have been established by identifying the Bergman space of the disk with the restriction of the Hardy space of the bidisk to the diagonal.”
On multiplication operators on the Bergman space: Similarity, unitary equivalence and reducing subspaces (2011, background): “As pointed out in [11], for each inner function φ which is not a Mobius map, the reducing subspaces of Mφ are in one to one corresponding with the closed subspaces of H2(T) φH2(T) [7],[12].”