Discrete spectrum (mathematics)

In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank of the corresponding Riesz projector is finite.

Definition

A point [math]\displaystyle{ \lambda\in\C }[/math] in the spectrum [math]\displaystyle{ \sigma(A) }[/math] of a closed linear operator [math]\displaystyle{ A:\,\mathfrak{B}\to\mathfrak{B} }[/math] in the Banach space [math]\displaystyle{ \mathfrak{B} }[/math] with domain [math]\displaystyle{ \mathfrak{D}(A)\subset\mathfrak{B} }[/math] is said to belong to discrete spectrum [math]\displaystyle{ \sigma_{\mathrm{disc}}(A) }[/math] of [math]\displaystyle{ A }[/math] if the following two conditions are satisfied:

1. [math]\displaystyle{ \lambda }[/math] is an isolated point in [math]\displaystyle{ \sigma(A) }[/math];
2. The rank of the corresponding Riesz projector [math]\displaystyle{ P_\lambda=\frac{-1}{2\pi\mathrm{i}}\oint_\Gamma(A-z I_{\mathfrak{B}})^{-1}\,dz }[/math] is finite.

Here [math]\displaystyle{ I_{\mathfrak{B}} }[/math] is the identity operator in the Banach space [math]\displaystyle{ \mathfrak{B} }[/math] and [math]\displaystyle{ \Gamma\subset\C }[/math] is a smooth simple closed counterclockwise-oriented curve bounding an open region [math]\displaystyle{ \Omega\subset\C }[/math] such that [math]\displaystyle{ \lambda }[/math] is the only point of the spectrum of [math]\displaystyle{ A }[/math] in the closure of [math]\displaystyle{ \Omega }[/math]; that is, [math]\displaystyle{ \sigma(A)\cap\overline{\Omega}=\{\lambda\}. }[/math]

Relation to normal eigenvalues

The discrete spectrum [math]\displaystyle{ \sigma_{\mathrm{disc}}(A) }[/math] coincides with the set of normal eigenvalues of [math]\displaystyle{ A }[/math]:

[math]\displaystyle{ \sigma_{\mathrm{disc}}(A)=\{\mbox{normal eigenvalues of }A\}. }[/math]

Relation to isolated eigenvalues of finite algebraic multiplicity

In general, the rank of the Riesz projector can be larger than the dimension of the root lineal [math]\displaystyle{ \mathfrak{L}_\lambda }[/math] of the corresponding eigenvalue, and in particular it is possible to have [math]\displaystyle{ \mathrm{dim}\,\mathfrak{L}_\lambda\lt \infty }[/math], [math]\displaystyle{ \mathrm{rank}\,P_\lambda=\infty }[/math]. So, there is the following inclusion:

[math]\displaystyle{ \sigma_{\mathrm{disc}}(A)\subset\{\mbox{isolated points of the spectrum of }A\mbox{ with finite algebraic multiplicity}\}. }[/math]

In particular, for a quasinilpotent operator

[math]\displaystyle{ Q:\,l^2(\N)\to l^2(\N),\qquad Q:\,(a_1,a_2,a_3,\dots)\mapsto (0,a_1/2,a_2/2^2,a_3/2^3,\dots), }[/math]

one has [math]\displaystyle{ \mathfrak{L}_\lambda(Q)=\{0\} }[/math], [math]\displaystyle{ \mathrm{rank}\,P_\lambda=\infty }[/math], [math]\displaystyle{ \sigma(Q)=\{0\} }[/math], [math]\displaystyle{ \sigma_{\mathrm{disc}}(Q)=\emptyset }[/math].

Relation to the point spectrum

The discrete spectrum [math]\displaystyle{ \sigma_{\mathrm{disc}}(A) }[/math] of an operator [math]\displaystyle{ A }[/math] is not to be confused with the point spectrum [math]\displaystyle{ \sigma_{\mathrm{p}}(A) }[/math], which is defined as the set of eigenvalues of [math]\displaystyle{ A }[/math]. While each point of the discrete spectrum belongs to the point spectrum,

[math]\displaystyle{ \sigma_{\mathrm{disc}}(A)\subset\sigma_{\mathrm{p}}(A), }[/math]

the converse is not necessarily true: the point spectrum does not necessarily consist of isolated points of the spectrum, as one can see from the example of the left shift operator, [math]\displaystyle{ L:\,l^2(\N)\to l^2(\N), \quad L:\,(a_1,a_2,a_3,\dots)\mapsto (a_2,a_3,a_4,\dots). }[/math] For this operator, the point spectrum is the unit disc of the complex plane, the spectrum is the closure of the unit disc, while the discrete spectrum is empty:

[math]\displaystyle{ \sigma_{\mathrm{p}}(L)=\mathbb{D}_1, \qquad \sigma(L)=\overline{\mathbb{D}_1}; \qquad \sigma_{\mathrm{disc}}(L)=\emptyset. }[/math]

See also

• Spectrum (functional analysis)
• Decomposition of spectrum (functional analysis)
• Normal eigenvalue
• Essential spectrum
• Spectrum of an operator
• Resolvent formalism
• Riesz projector
• Fredholm operator
• Operator theory