Abstract
The main result of the paper shows that, for 1 < p < ∞ and 1 ≤ q < ∞, a linear operator T: ℓ p → ℓ q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p = 1). For 1 < p ≠ q < ∞, as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T: ℓ p → ℓ q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p = q). We also investigate lineability of the sets of norm-attaining and non-norm attaining operators.
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Partially supported by INCT-Matemática, CNPq Grants nos. 620108/2008-8 (Ed. Casadinho), 471686/2008-5 (Ed. Universal), 308084/2006-3 and PROCAD NF-Capes
Partially supported by CNPq-Brazil and by NSF grant DMS 0600930.
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Pellegrino, D., Teixeira, E.V. Norm optimization problem for linear operators in classical Banach spaces. Bull Braz Math Soc, New Series 40, 417–431 (2009). https://doi.org/10.1007/s00574-009-0019-7
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DOI: https://doi.org/10.1007/s00574-009-0019-7