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We show that a recent construction of V. Müller (|4|) gives a pair of commuting Banach space operators for which the Harte spectrum is not contained in the Taylor spectrum.
The algebra B(ℋ) of all bounded operators on a Hilbert space ℋ is generated in the strong operator topology by a single one-dimensional projection and a family of commuting unitary operators with cardinality not exceeding dim ℋ. This answers Problem 8 posed by W. Żelazko in [6].
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