On normal operator logarithms
Let X,Y be normal bounded operators on a Hilbert space such that e X=eY. If the spectra of X and Y are contained in the strip S of the complex plane defined by |I(z)|≤π, we show that |X|=|Y|. If Y is only assumed to be bounded, then |X|Y=Y|X|. We give a formula for X-Y in terms of spectral projectio...
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2013 |
| País: | Argentina |
| Recursos: | Universidad Nacional de La Plata |
| Repositorio: | SEDICI (UNLP) |
| Idioma: | inglés |
| OAI Identifier: | oai:sedici.unlp.edu.ar:10915/85121 |
| Acesso em linha: | http://sedici.unlp.edu.ar/handle/10915/85121 |
| Access Level: | acceso abierto |
| Palavra-chave: | Matemática Exponential map Normal operator Spectral theorem |
| Resumo: | Let X,Y be normal bounded operators on a Hilbert space such that e X=eY. If the spectra of X and Y are contained in the strip S of the complex plane defined by |I(z)|≤π, we show that |X|=|Y|. If Y is only assumed to be bounded, then |X|Y=Y|X|. We give a formula for X-Y in terms of spectral projections of X and Y provided that X,Y are normal and e X=eY. If X is an unbounded self-adjoint operator, which does not have (2k+1)π,k∈ℤ, as eigenvalues, and Y is normal with spectrum in S satisfying eiX=eY, then Y∈{ e iX}″. We give alternative proofs and generalizations of results on normal operator exponentials proved by Schmoeger. |
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