On normal operator logarithms
Let X,Y be normal bounded operators on a Hilbert space such that e^X=e^Y. If the spectra of X and Y are contained in the strip S of the complex plane defined by |Im(z)|leq pi, 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 p...
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2013 |
| País: | Argentina |
| Recursos: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositório: | CONICET Digital (CONICET) |
| Idioma: | inglês |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/3374 |
| Acesso em linha: | http://hdl.handle.net/11336/3374 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Exponential Map Normal Operator Spectral Theorem https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Resumo: | Let X,Y be normal bounded operators on a Hilbert space such that e^X=e^Y. If the spectra of X and Y are contained in the strip S of the complex plane defined by |Im(z)|leq pi, 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=e^Y. <br />If X is an unbounded self-adjoint operator, which does not have (2k+1) pi, k in Z, as eigenvalues, and Y is normal with spectrum in S satisfying e^{iX}=e^Y, then Y in { e^{iX} }´´. We give alternative proofs and generalizations of results on normal operator exponentials proved by Ch. Schmoeger. |
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