Bounds on nonlinear operators in finite-dimensional banach spaces |
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Authors: | Gustaf Söderlind |
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Institution: | (1) Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-10044 Stockholm, Sweden |
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Abstract: | Summary We consider Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces. Boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals. These functionals are used to assess qualitative properties such as invertibility, and also enable a generalization of some well-known matrix results directly to nonlinear operators. Closely related to the numerical range of a matrix, the Gerschgorin domain is introduced for nonlinear operators. This point set in the complex plane is always convex and contains the spectrum of the operator's Jacobian matrices. Finally, we focus on nonlinear operators in Hilbert space and hint at some generalizations of the von Neumann spectral theory. |
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Keywords: | AMS(MOS) 47H05 47H12 65H10 65L07 CR: G1 5 G1 7 |
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