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On the Functional Equation $\boldsymbol{U}_{\boldsymbol{t}}\boldsymbol{+} \boldsymbol{U}_{\boldsymbol{-t}} = \boldsymbol{V}_{\boldsymbol{t}} \boldsymbol{+} \boldsymbol{V}_{\boldsymbol{-t}}$ in a Banach Space
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