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In the following text we compute the adjoint of weighted generalized shift operators over Hilbert spaces. We show for a conjugate invariant subset A of C, the additive semigroup generated by A\0\-weighted generalized shifts over Hilbert space H is adjoint invariant if and only if H is a finite dimensional Hilbert space or 0 is not a limit point of A.
Shirazi et al. (Sat,) studied this question.
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