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For ''λ'' ∈ '''C''' with |''λ''| < 1,
<math display="block">x = (1, \lambda, \lambda ^2, \dots) \in l^p</math>
and ''T x'' = ''λ x''. Consequently, the point spectrum of ''T'' contains the open unit disk. Now, ''T*'' has no eigenvalues, i.e. ''σ<sub>p</sub>''(''T*'') is empty. Thus, invoking reflexivity and the theorem
The spectrum of a bounded operator is closed, which implies the unit circle, { |''λ''| = 1 } ⊂ '''C''', is in ''σ''(''T''). Again by reflexivity of ''l <sup>p</sup>'' and the theorem given above (this time, that {{math|''σ<sub>r</sub>''(''T'') ⊂ ''σ<sub>p</sub>''(''T''*)}}), we have that ''σ<sub>r</sub>''(''T'') is also empty. Therefore, for a complex number ''λ'' with unit norm, one must have ''λ'' ∈ ''σ<sub>p</sub>''(''T'') or ''λ'' ∈ ''σ<sub>c</sub>''(''T''). Now if |''λ''| = 1 and
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