Square variation norm and non-negative, non-increasing sequences - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:03:03Z http://mathoverflow.net/feeds/question/68173 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68173/square-variation-norm-and-non-negative-non-increasing-sequences Square variation norm and non-negative, non-increasing sequences Sellapan Nathan 2011-06-18T22:58:46Z 2011-06-19T00:18:21Z <p>I am trying to understand the properties of square variation, namely, the possibility of preserving it under certain operations. I am following Albiac &amp; Kalton's book: Let $J$ stand for the usual definition of the James space (see Definition 3.4.1 p. 62).</p> <p>Let $\mathcal{P}$ be the family of all non-increasing sequences of non-negative real numbers, convergent to 0.</p> <p>Suppose that we are given a real function $f$ on positive integers taking non-negative values only. Assume, moreover, that:</p> <ul> <li><p>$f(0)=0$</p></li> <li><p>$(f(x_k))_{k=1}^\infty \in J\cap \mathcal{P}$ </p></li> <li><p>$\sum_{k=1}^n f(x_k+y_k)\leq \sum_{k=1}^n f(x_k)+\sum_{k=1}^n f(y_k)$ </p></li> </ul> <p>for every $(x_k)_{k=1}^\infty$ and $(y_k)_{k=1}^\infty$ in $c_0$.</p> <p>Does the inequality hold $$\|(f(x_k+y_k))_{k=1}^\infty\|_J\leq \|(f(x_k))_{k=1}^\infty\|_J + \|(g(x_k))_{k=1}^\infty\|_J$$</p> <p>In particular, must $(f(x_k+y_k))_{k=1}^\infty\in J$?</p> <p>If the answer is negative, are there any sufficient conditions for $f$ to satisfy this inequality?</p>