Talk:Borel functional calculus

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Mathematics rating:	Start Class	Mid Priority	Field: Analysis
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 22:41, 15 June 2007 (UTC)

CSTAR, good edit. Thanks. Mct mht 05:48, 7 April 2006 (UTC)

[edit] Complex valued functions

By considering Borel functions on the complex plane, the requirement of the operator T being self-adjoint will be changed to being normal.

the normal case is more general anyhow, but there's no real-valued restriction for the self-adjoint case. we simply consider the real and imaginary parts of a function of a real variable. given an operator T, the Borel calculus gives a spectral measure, or the resolution of the identity, {E_B} indexed by the Borel sets in σ(T). we define f(T) by <h, f(T) h> = ∫ f d <h, E_B(T) h>. f can be complex-valued. Mct mht 14:55, 16 August 2006 (UTC)

in the special case that f is real valued, f(T) would also be self adjoint. Mct mht 14:58, 16 August 2006 (UTC)

[edit] Domain of functional calculus

I don't believe the most recent edit concerning the domain of the functional calculus is correct. The functional calculus structly speaking is a function of 2 arguments a function h and an operator T.

• For the continuous functional calculus, h varies over continuous functions and T over self-adjoint operators (or maybe normal ones depending)
• For the Borel functional calculus, h varies over Borel functions and T over self-adjoint operators (or maybe normal ones depending)

What sense does it make to say the domain of the calculus is an algebra of operators?

Perhaps what the article means to say is that for a fixed T, the range of the funcal calculus is a C*-algebra or a von Neuman algebra.--CSTAR 14:13, 13 December 2006 (UTC)

i was thinking of C(σ(T)) and L^∞(σ(T)), the domains. and yes i took "functional calculus" to mean the map that assigns f to f(T), for a fixed T. in that case, seems to me that the domain/range wording is not an issue. Mct mht 17:40, 13 December 2006 (UTC)