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Decomposition of spectrum (functional analysis): Difference between revisions

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m Capitalising short description "construction in functional analysis, useful to solve differential equations" per WP:SDFORMAT (via Bandersnatch)
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This subsection briefly sketches the development of this calculus. The idea is to first establish the continuous functional calculus then pass to measurable functions via the [[Riesz representation theorem|Riesz-Markov representation theorem]]. For the continuous functional calculus, the key ingredients are the following:
 
:1.# If ''T'' is self-adjoint, then for any polynomial ''P'', the operator norm satisfies <math display="block">\| P(T) \| = \sup_{\lambda \in \sigma(T)} |P(\lambda)|.</math>
:2.# The [[Stone–Weierstrass theorem]], which implies that the family of polynomials (with complex coefficients), is dense in ''C''(''&sigma;σ''(''T'')), the continuous functions on ''&sigma;σ''(''T'').▼
 
::<math>\| P(T) \| = \sup_{\lambda \in \sigma(T)} |P(\lambda)|.</math>
▲:2. The [[Stone–Weierstrass theorem]], which implies that the family of polynomials (with complex coefficients), is dense in ''C''(''&sigma;''(''T'')), the continuous functions on ''&sigma;''(''T'').
 
The family ''C''(''σ''(''T'')) is a [[Banach algebra]] when endowed with the uniform norm. So the mapping
:<math display="block">P \rightarrow P(T)</math>▼
 
▲:<math>P \rightarrow P(T)</math>
 
is an isometric homomorphism from a dense subset of ''C''(''σ''(''T'')) to ''B''(''H''). Extending the mapping by continuity gives ''f''(''T'') for ''f'' ∈ C(''σ''(''T'')): let ''P<sub>n</sub>'' be polynomials such that ''P<sub>n</sub>'' → ''f'' uniformly and define ''f''(''T'') = lim ''P<sub>n</sub>''(''T''). This is the continuous functional calculus.
 
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