Abstract.
It is shown that if 1 < p < ∞ and X is a subspace or a quotient of an ℓ p -direct sum of finite dimensional Banach spaces, then for any compact operator T on X such that ∥I + T∥ > 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that ∥I + T∥ > 1 and I + T does not attain its norm.
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The author would like to thank E. Shargorodsky for his interest and comments.
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Shkarin, S. Norm Attaining Operators and Pseudospectrum. Integr. equ. oper. theory 64, 115–136 (2009). https://doi.org/10.1007/s00020-009-1676-z
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DOI: https://doi.org/10.1007/s00020-009-1676-z