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A unilateral weighted shift A is said to be simple if its weight sequence {αn} satisfies ∇ 3 (α 2 n ) = 0 for all n 2. We prove that if A and B are two simple unilateral weighted shifts, then A ⊗ I + I ⊗ B is reducible if and only if A and B are unitarily equivalent. We also study the reducing subspaces of A k ⊗ I + I ⊗ B l and give some examples. As an application, we study the reducing subspaces of multiplication operators M z k +αw l on function spaces.

Polydimethylsiloxane (PDMS) and polyurethane acrylate (PUA) are excellent pattern transfer materials. In this study, PDMS-PUA bi-directional replication technology is explored using the PDMS grating as a template, and relevant technical issues are discussed in detail. Special surface treatment and process optimization are applied to solve the problems of demolding, PDMS polymerization inhibition, and substrate flatness. Further experiments show that the technology can be employed to replicate nanoscale structures and has the potential value of prolonging the longevity of the original template. Additionally, utilizing the advantage of the high elasticity of PDMS materials, two applications of bi-directional replication technology are demonstrated. One is to increase the line-density of the grating by stretching, and the experimental results show that the line-density of the grating increased by 26.6%. The other one is to fabricate the convex grating. Compared with the original planar PDMS grating, the resolution of the first-order diffraction spectrum of the convex grating at the focal point has been greatly improved. Since this technology requires simple equipment, and PDMS and PUA are reusable, it has the advantages of low cost, simplicity, and rapid fabrication. The two application examples also indicate that the technology has good application value.