Editing Jordan normal form (section)

== Compact operators ==
A result analogous to the Jordan normal form holds for [[compact operator]]s on a [[Banach space]]. One restricts to compact operators because every point ''x'' in the spectrum of a compact operator ''T'' is an eigenvalue; The only exception is when ''x'' is the limit point of the spectrum. This is not true for bounded operators in general. To give some idea of this generalization, we first reformulate the Jordan decomposition in the language of [[functional analysis]].

=== 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.

=== The finite-dimensional case ===

In the finite-dimensional case, ''σ''(''T'') = {''λ''<sub>''i''</sub>} is a finite discrete set in the complex plane. Let ''e''<sub>''i''</sub> be the function that is 1 in some open neighborhood of ''λ''<sub>''i''</sub> and 0 elsewhere. By property 3 of the functional calculus, the operator

:<math>e_i(T)</math>

is a projection. Moreover, let ''ν<sub>i</sub>'' be the index of ''λ''<sub>''i''</sub> and

:<math>f(z)= (z - \lambda_i)^{\nu_i}.</math>

The spectral mapping theorem tells us

:<math> f(T) e_i (T) = (T - \lambda_i)^{\nu_i} e_i (T)</math>

has spectrum {0}. By property 1, ''f''(''T'') can be directly computed in the Jordan form, and by inspection, we see that the operator ''f''(''T'')''e<sub>i</sub>''(''T'') is the zero matrix.

By property 3, ''f''(''T'') ''e''<sub>''i''</sub>(''T'') = ''e''<sub>''i''</sub>(''T'') ''f''(''T''). So ''e''<sub>''i''</sub>(''T'') is precisely the projection onto the subspace

:<math>\operatorname{Ran} e_i (T) = \ker(T - \lambda_i)^{\nu_i}.</math>

The relation

:<math>\sum_i e_i = 1</math>

implies

:<math>\mathbb{C}^n = \bigoplus_i \; \operatorname{Ran} e_i (T) = \bigoplus_i \ker(T - \lambda_i)^{\nu_i}</math>

where the index ''i'' runs through the distinct eigenvalues of ''T''. This is the invariant subspace decomposition

:<math>\mathbb{C}^n = \bigoplus_i Y_i</math>

given in a previous section. Each ''e<sub>i</sub>''(''T'') is the projection onto the subspace spanned by the Jordan chains corresponding to ''λ''<sub>''i''</sub> and along the subspaces spanned by the Jordan chains corresponding to v<sub>''j''</sub> for ''j'' ≠ ''i''. In other words, ''e<sub>i</sub>''(''T'') = ''P''(''λ''<sub>''i''</sub>;''T''). This explicit identification of the operators ''e<sub>i</sub>''(''T'') in turn gives an explicit form of holomorphic functional calculus for matrices:

:For all ''f'' ∈ Hol(''T''),

:<math>f(T) = \sum_{\lambda_i \in \sigma(T)} \sum_{k = 0}^{\nu_i -1} \frac{f^{(k)}}{k!} (T - \lambda_i)^k e_i (T).</math>

Notice that the expression of ''f''(''T'') is a finite sum because, on each neighborhood of v<sub>''i''</sub>, we have chosen the [[Taylor series]] expansion of ''f'' centered at v<sub>''i''</sub>.

=== Poles of an operator ===

Let ''T'' be a bounded operator ''λ'' be an isolated point of ''σ''(''T''). (As stated above, when ''T'' is compact, every point in its spectrum is an isolated point, except possibly the limit point 0.)

The point ''λ'' is called a '''pole''' of operator ''T'' with order ''ν'' if the [[Resolvent formalism|resolvent]] function ''R''<sub>''T''</sub> defined by

:<math> R_T(\lambda) = (\lambda - T)^{-1}</math>

has a [[pole (complex analysis)|pole]] of order ''ν'' at ''λ''.

We will show that, in the finite-dimensional case, the order of an eigenvalue coincides with its index. The result also holds for compact operators.

Consider the annular region ''A'' centered at the eigenvalue ''λ'' with sufficiently small radius ''ε'' such that the intersection of the open disc ''B<sub>ε</sub>''(''λ'') and ''σ''(''T'') is {''λ''}. The resolvent function ''R''<sub>''T''</sub> is holomorphic on ''A''.
Extending a result from classical function theory, ''R''<sub>''T''</sub> has a [[Laurent series]] representation on ''A'':

:<math>R_T(z) = \sum_{-\infty}^\infty a_m (\lambda - z)^m</math>

where

:<math>a_{-m} = - \frac{1}{2 \pi i} \int_C (\lambda - z) ^{m-1} (z - T)^{-1} d z</math> and ''C'' is a small circle centered at&nbsp;''λ''.

By the previous discussion on the functional calculus,

:<math> a_{-m} = -(\lambda - T)^{m-1} e_\lambda (T)</math> where <math> e_\lambda</math> is 1 on <math> B_\varepsilon(\lambda)</math> and 0 elsewhere.

But we have shown that the smallest positive integer ''m'' such that

:<math>a_{-m} \neq 0</math> and <math>a_{-l} = 0 \; \; \forall \; l \geq m</math>

is precisely the index of ''λ'', ''ν''(''λ''). In other words, the function ''R''<sub>''T''</sub> has a pole of order ''ν''(''λ'') at ''λ''.