What is the "correct" generalization of operator norms for nonlinear operators? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T23:35:06Z http://mathoverflow.net/feeds/question/60823 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60823/what-is-the-correct-generalization-of-operator-norms-for-nonlinear-operators What is the "correct" generalization of operator norms for nonlinear operators? suVRit 2011-04-06T14:37:54Z 2011-04-06T18:05:36Z <p>I have been recently wondering what is a (or even the) "correct" generalization of the notion of an <em>operator norm</em> to nonlinear operators? </p> <p>Please excuse the naivete of my question; if you think that question will benefit from being made more precise, then I will appreciate help towards making it so.</p> <hr> <p>Because I lack formal education in mathematics, I might be missing something obvious or well-known here. Could somebody point me in the right direction, and let me know what are the key concepts to think about when defining operator norms for nonlinear operators?</p> <p>Some vague ideas that occurred to me:</p> <ol> <li><p>Linearizing the operator (locally), so the essentially traditional operator norms of the linearized operator could be considered? This sounds very unsatisfactory though.</p></li> <li><p>If \$A\$ is a nonlinear operator for which we can sensibly define \$\log A\$, maybe that helps in tackling the nonlinearity.</p></li> </ol> http://mathoverflow.net/questions/60823/what-is-the-correct-generalization-of-operator-norms-for-nonlinear-operators/60826#60826 Answer by András Bátkai for What is the "correct" generalization of operator norms for nonlinear operators? András Bátkai 2011-04-06T15:29:55Z 2011-04-06T18:05:36Z <p>Probably the answer depends on the context, where you need the generalized concept.</p> <p>Unfortunately, usually the linearized operator is not bounded anymore, because it contains differential operators.</p> <p>As pointed out by Mikael, the <a href="http://en.wikipedia.org/wiki/Lipschitz_continuity" rel="nofollow">Lipschitz constant</a> may be one possibility. For example, in the <a href="http://de.wikipedia.org/wiki/Michael_Crandall" rel="nofollow">Crandall</a>-<a href="http://en.wikipedia.org/wiki/Thomas_M._Liggett" rel="nofollow">Liggett</a> theory of nonlinear semigroups the <a href="http://en.wikipedia.org/wiki/Hille-Yosida_theorem" rel="nofollow">Hille-Yosida</a> generation theorem on linear contraction semigroups is generalized to nonlinear contractions. Here, definitely the Lipschitz constant replaces the role of the operator norm. Or in the <a href="http://en.wikipedia.org/wiki/Banach_fixed_point_theorem" rel="nofollow">Banach fixed point theorem</a> the convergence of the geometric series is generalized to iterations of nonlinear maps.</p> <p>But there might be other answers, I am really curious.</p>