Norm optimization problem for linear operators in classical Banach spaces
Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society - Tập 40 - Trang 417-431 - 2009
Tóm tắt
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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