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I have been recently wondering what is a (or even the) "correct" generalization of the notion of an operator norm to nonlinear operators?

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.


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?

Some vague ideas that occurred to me:

  1. Linearizing the operator (locally), so the essentially traditional operator norms of the linearized operator could be considered? This sounds very unsatisfactory though.

  2. If $A$ is a nonlinear operator for which we can sensibly define $\log A$, maybe that helps in tackling the nonlinearity.

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2  
Maybe the Lispschitz constant? –  Mikael de la Salle Apr 6 '11 at 14:41
    
Thanks Mikael; I guess linearizing takes us to Lipschitz? But for functions that are only locally Lipschitz? or which flavors of Lipschitz? Perhaps you'd care to expand a bit on your comment? –  Suvrit Apr 6 '11 at 15:36
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A useful bit of perspective on this question is provided by recalling that for a linear operator $A$ between normed spaces, the following are all equivalent: $A$ is continuous, $A$ is uniformly continuous, $A$ is Lipschitz, $\| A \|$ is finite. Without linearity the equivalences all fail. As abatkai points out, how to generalize $\| A \|$ appropriately depends on the context, in particular which properties of $A$ are actually important for you. –  Mark Meckes Apr 6 '11 at 15:38
    
Usually a norm is used to turn a vector space into a normed space. Do you have a special vector space of nonlinear operators in mind? Probably the space of all nonlinear operators between two (normed?) spaces is too large... –  Dirk Apr 6 '11 at 19:39
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Look at the book "Geometric nonlinear functional analysis" by Benyamini and Lindenstrauss. –  Bill Johnson Apr 7 '11 at 13:33
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1 Answer

Probably the answer depends on the context, where you need the generalized concept.

Unfortunately, usually the linearized operator is not bounded anymore, because it contains differential operators.

As pointed out by Mikael, the Lipschitz constant may be one possibility. For example, in the Crandall-Liggett theory of nonlinear semigroups the Hille-Yosida 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 Banach fixed point theorem the convergence of the geometric series is generalized to iterations of nonlinear maps.

But there might be other answers, I am really curious.

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