Editing Jordan normal form (section)

=== Holomorphic functional calculus ===
{{ Further|holomorphic functional calculus}}
Let ''X'' be a Banach space, ''L''(''X'') be the bounded operators on ''X'', and ''σ''(''T'') denote the [[spectrum (functional analysis)|spectrum]] of ''T'' ∈ ''L''(''X''). The [[holomorphic functional calculus]] is defined as follows:

Fix a bounded operator ''T''. Consider the family Hol(''T'') of complex functions that is [[Holomorphic function|holomorphic]] on some open set ''G'' containing ''σ''(''T''). Let Γ = {''γ<sub>i</sub>''} be a finite collection of [[Jordan curve]]s such that ''σ''(''T'') lies in the ''inside'' of Γ, we define ''f''(''T'') by

: <math>f(T) = \frac 1 {2 \pi i} \int_\Gamma f(z)(z - T)^{-1} \, dz.</math>

The open set ''G'' could vary with ''f'' and need not be connected. The integral is defined as the limit of the Riemann sums, as in the scalar case. Although the integral makes sense for continuous ''f'', we restrict to holomorphic functions to apply the machinery from classical function theory (for example, the Cauchy integral formula). The assumption that ''σ''(''T'') lie in the inside of Γ ensures ''f''(''T'') is well defined; it does not depend on the choice of Γ. The functional calculus is the mapping Φ from Hol(''T'') to ''L''(''X'') given by

: <math>\; \Phi(f) = f(T).</math>

We will require the following properties of this functional calculus:
# Φ extends the polynomial functional calculus.
# The ''spectral mapping theorem'' holds: ''σ''(''f''(''T'')) = ''f''(''σ''(''T'')).
# Φ is an algebra homomorphism.