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Additional resources for Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space
Proofs of lemma 5C and of proposition 3C: as for lenma 5B and 3B. • Again, a basis comoatibl~ with such as e in proposition 4C is said to be _h. / to proposition classical 3C, XJ~ = JQX I. operator x, then proposition I = ~X ~ U ( ~ ) be as in proposition and let Fur thermore, Sp(~Q) 3 : Let h be a Caftan subalgebra complex Lie algebra c~ finite rank operators. of a Then h is equal to its normalizer. Proof : Remark. SI(~ Caftan immediate. • Let ~ ; CO) , be one of the infinite dimensional o(~ s/balgebra 'JR; Co) of ~ N(H) = ~X ~ a ] [H,X] = O~ h SD(~ ald let trivially from proposition particular, and ,JQ; Co).
Of ~ is m a x i m a l all its elements it follows which over Is among the are semi-simple. that Caftan subalgebras 4 are Cartan sub algebras of ~ of in the sense. Lie algebra sense. Let h let and let Then invariant h ~ h be a finite be a Caftan there exists a c-involution 11, & dimensional be a finite (Kostant th~or~mes subalgebra ),there is invariant. din~nsional subslgebra (, expos~ be a Caftan standard which of ~ to definition Conversely, let c~ this d e f i n i t i o n th~ n~tter of this chapter a Caftan subalgebra sub al gebras ~ usual so that be a semi-simple the abelian In particular, When to chapter if A critic of semi-simplicity subalgebra among abelian ideals.
Rs,@ = @ - l c @ let . all on a f i n i t e d i m e n s i o n a l abcve, s @RC s @~@ ; ~ rs,~ @r sense be as z Conversely, Then ~sual c-involution ~ ~  i n the Let of ~ with ~ X~ = - X 1 . conjugation a convenient ~ a n d its for ~l Lie to the c o m p a c t f o r m Standard results for involutive -~ -X* u = IX ~ ~ A conjugation = (a(X))~ ~-isomorphlc is the map Lie a l g e b r a , such that and is a involutive X ; ~ X• . ; [o(x), ~ complexification o__ff ~ a complex c : ~ a([x,Y]) form be d e n o t e d by c2(X) A real ~ e ~ be and the ~he map "-automorphism ~-automorphism involutive form of and for all X , Y ~ _s.