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Understanding Gillman's proof of the Chernoff bound for expander graphs

My question is about the proof of Claim 1 in this paper: Gillman (1993). At the end of the proof, the author says: The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, …