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.

A point ${\displaystyle \lambda \in \mathbb {C} }${\displaystyle \lambda \in \mathbb {C} } in the spectrum ${\displaystyle \sigma (A)}${\displaystyle \sigma (A)} of a closed linear operator ${\displaystyle A:,円{\mathfrak {B}}\to {\mathfrak {B}}}${\displaystyle A:,円{\mathfrak {B}}\to {\mathfrak {B}}} in the Banach space ${\displaystyle {\mathfrak {B}}}${\displaystyle {\mathfrak {B}}} with domain ${\displaystyle {\mathfrak {D}}(A)\subset {\mathfrak {B}}}${\displaystyle {\mathfrak {D}}(A)\subset {\mathfrak {B}}} is said to belong to discrete spectrum ${\displaystyle \sigma _{\mathrm {disc} }(A)}${\displaystyle \sigma _{\mathrm {disc} }(A)} of ${\displaystyle A}${\displaystyle A} if the following two conditions are satisfied:^[1]

1. ${\displaystyle \lambda }${\displaystyle \lambda } is an isolated point in ${\displaystyle \sigma (A)}${\displaystyle \sigma (A)};
2. The rank of the corresponding Riesz projector ${\displaystyle P_{\lambda }={\frac {-1}{2\pi \mathrm {i} }}\oint _{\Gamma }(A-zI_{\mathfrak {B}})^{-1},円dz}${\displaystyle P_{\lambda }={\frac {-1}{2\pi \mathrm {i} }}\oint _{\Gamma }(A-zI_{\mathfrak {B}})^{-1},円dz} is finite.

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

The discrete spectrum ${\displaystyle \sigma _{\mathrm {disc} }(A)}${\displaystyle \sigma _{\mathrm {disc} }(A)} coincides with the set of normal eigenvalues of ${\displaystyle A}${\displaystyle A}:

${\displaystyle \sigma _{\mathrm {disc} }(A)=\{{\mbox{normal eigenvalues of }}A\}.}${\displaystyle \sigma _{\mathrm {disc} }(A)=\{{\mbox{normal eigenvalues of }}A\}.}^{[2] [3] [4]}

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

${\displaystyle \sigma _{\mathrm {disc} }(A)\subset \{{\mbox{isolated points of the spectrum of }}A{\mbox{ with finite algebraic multiplicity}}\}.}${\displaystyle \sigma _{\mathrm {disc} }(A)\subset \{{\mbox{isolated points of the spectrum of }}A{\mbox{ with finite algebraic multiplicity}}\}.}

In particular, for a quasinilpotent operator

${\displaystyle Q:,円l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} ),\qquad Q:,円(a_{1},a_{2},a_{3},\dots )\mapsto (0,a_{1}/2,a_{2}/2^{2},a_{3}/2^{3},\dots ),}${\displaystyle Q:,円l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} ),\qquad Q:,円(a_{1},a_{2},a_{3},\dots )\mapsto (0,a_{1}/2,a_{2}/2^{2},a_{3}/2^{3},\dots ),}

one has ${\displaystyle {\mathfrak {L}}_{\lambda }(Q)=\{0\}}${\displaystyle {\mathfrak {L}}_{\lambda }(Q)=\{0\}}, ${\displaystyle \mathrm {rank} ,円P_{\lambda }=\infty }${\displaystyle \mathrm {rank} ,円P_{\lambda }=\infty }, ${\displaystyle \sigma (Q)=\{0\}}${\displaystyle \sigma (Q)=\{0\}}, ${\displaystyle \sigma _{\mathrm {disc} }(Q)=\emptyset }${\displaystyle \sigma _{\mathrm {disc} }(Q)=\emptyset }.

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

${\displaystyle \sigma _{\mathrm {disc} }(A)\subset \sigma _{\mathrm {p} }(A),}${\displaystyle \sigma _{\mathrm {disc} }(A)\subset \sigma _{\mathrm {p} }(A),}

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, ${\displaystyle L:,円l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} ),\quad L:,円(a_{1},a_{2},a_{3},\dots )\mapsto (a_{2},a_{3},a_{4},\dots ).}${\displaystyle L:,円l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} ),\quad L:,円(a_{1},a_{2},a_{3},\dots )\mapsto (a_{2},a_{3},a_{4},\dots ).} 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:

${\displaystyle \sigma _{\mathrm {p} }(L)=\mathbb {D} _{1},\qquad \sigma (L)={\overline {\mathbb {D} _{1}}};\qquad \sigma _{\mathrm {disc} }(L)=\emptyset .}${\displaystyle \sigma _{\mathrm {p} }(L)=\mathbb {D} _{1},\qquad \sigma (L)={\overline {\mathbb {D} _{1}}};\qquad \sigma _{\mathrm {disc} }(L)=\emptyset .}

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

• Spectral theory