Norm optimization problem for linear operators in classical Banach spaces

摘要

The main result of the paper shows that, for 1 < p < a and 1 a parts per thousand currency sign q < a, a linear operator T: a"" (p) -> a"" (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 not equal q < a, as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T: a"" (p) -> a"" (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.