Douglas' lemma

In operator theory, an area of mathematics, Douglas' lemma relates factorization, range inclusion, and majorization of Hilbert space operators. It is generally attributed to Ronald G. Douglas, although Douglas acknowledges that aspects of the result may already have been known. The statement of the result is as follows:

Theorem: If A and B are bounded operators on a Hilbert space H, the following are equivalent:

1. $\text{im}(A)\subseteq\text{im}(B).\,$
2. $AA^*\leq\lambda^2 BB^*$ for some $\lambda\geq 0.\,$
3. There exists a bounded operator C on H such that A = BC.

Moreover, if these equivalent conditions hold, then there is a unique operator C such that

• $\Vert C \Vert^2=\inf\{\mu :\,AA^*\leq\mu BB^*\}.$
• ker(A) = ker(C)
• $\text{im}(C)\subseteq\overline{\text{im}(B^*)}$

See also

Positive operator

References

• Douglas, R.G.: "On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space". Proceedings of the American Mathematical Society 17, 413–415 (1966)

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.v · Categories: Mathematical analysis stubs Operator theory Wikimedia Foundation. 2010. Игры ⚽ Поможем написать курсовую Dougie Young Arcuate line of rectus sheath Look at other dictionaries: Hugh Douglas Hamilton — (c. 1734/1739 ndash; February 10, 1808) was an Irish artist during the mid to late 18th century. Hamilton was born in Dublin, Ireland around 1724, where he would study art under Robert West at the Dublin Society House concentrating in crayon and… …   Wikipedia Polar decomposition — In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to the polar form of a nonzero complex number z where r is the absolute value of z (a… …   Wikipedia Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… …   Wikipedia Infinite Monkey Theorem — Durch zufälliges Tippen von unendlicher Dauer auf einer Schreibmaschine werden mit Sicherheit alle Texte Shakespeares oder diverser Nationalbibliotheken entstehen. Das Infinite Monkey Theorem (v. engl. infinite „unendlich“; monkey „Affe“; theorem …   Deutsch Wikipedia Infinite monkey theorem — Durch zufälliges Tippen von unendlicher Dauer auf einer Schreibmaschine werden mit Sicherheit alle Texte Shakespeares oder diverser Nationalbibliotheken entstehen. Das Infinite Monkey Theorem (v. engl. infinite „unendlich“; monkey „Affe“; theorem …   Deutsch Wikipedia Satz vom Affen — Durch zufälliges Tippen von unendlicher Dauer auf einer Schreibmaschine werden mit Sicherheit alle Texte Shakespeares oder diverser Nationalbibliotheken entstehen. Das Infinite Monkey Theorem (v. engl. infinite „unendlich“; monkey „Affe“; theorem …   Deutsch Wikipedia Theorem der endlos tippenden Affen — Durch zufälliges Tippen von unendlicher Dauer auf einer Schreibmaschine werden mit Sicherheit alle Texte Shakespeares oder diverser Nationalbibliotheken entstehen. Das Infinite Monkey Theorem (v. engl. infinite „unendlich“; monkey „Affe“; theorem …   Deutsch Wikipedia Unendlich-viele-Affen-Theorem — Durch zufälliges Tippen von unendlicher Dauer auf einer Schreibmaschine werden mit Sicherheit alle Texte Shakespeares oder diverser Nationalbibliotheken entstehen. Das Infinite Monkey Theorem (v. engl. infinite „unendlich“; monkey „Affe“; theorem …   Deutsch Wikipedia Unendlich-viele-Affen-Theorie — Durch zufälliges Tippen von unendlicher Dauer auf einer Schreibmaschine werden mit Sicherheit alle Texte Shakespeares oder diverser Nationalbibliotheken entstehen. Das Infinite Monkey Theorem (v. engl. infinite „unendlich“; monkey „Affe“; theorem …   Deutsch Wikipedia Gödel's incompleteness theorems — In mathematical logic, Gödel s incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. The theorems are of… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”