Some mapping properties of \(p\)-summing adjoint operators
A sequence \(x_n\) in a Banach space \(X\) is called almost unconditionally summable if the series \(\sum \pm x_n\) converge for almost all choices of signs \(\pm\). Let \(X, Y\) be Banach spaces, \(u:X \to Y\) be a continuous linear operator. If \(u^*: Y^* \to X^*\) is \(p\)-summing for some \(p \geq 1\), then for any \(q \geq 2\), \(u\) takes almost unconditionally summable sequences in \(X\) into elements of \(l_q \widehat{\otimes} Y\), the projective product of \(l_q\) and \(Y\). Moreover, if \(X\) has type 2, then \(u\) takes almost unconditionally summable sequences in \(X\) into elements of \(l_2 \widehat{\otimes} Y\) if and only if \(u^*\) is 2-summing.
almost unconditionally summable sequence
scientific article; zbMATH DE number 2153767
scientific article