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 formula_1
in the spectrum formula_2 of a closed linear operator formula_3 in the Banach space formula_4 with domain formula_5 is said to belong to "discrete spectrum" formula_6 of formula_7 if the following two conditions are satisfied:
Here formula_11 is the identity operator in the Banach space formula_4 and formula_13 is a smooth simple closed counterclockwise-oriented curve bounding an open region formula_14 such that formula_8 is the only point of the spectrum of formula_7 in the closure of formula_17; that is, formula_18
Relation to normal eigenvalues.
The discrete spectrum formula_6 coincides with the set of normal eigenvalues of formula_7:
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 formula_22 of the corresponding eigenvalue, and in particular it is possible to have formula_23, formula_24. So, there is the following inclusion:
In particular, for a quasinilpotent operator
one has
formula_27, formula_24,
formula_29,
formula_30.
Relation to the point spectrum.
The discrete spectrum formula_6 of an operator formula_7 is not to be confused with the point spectrum formula_33, which is defined as the set of eigenvalues of formula_7.
While each point of the discrete spectrum belongs to the point spectrum,
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",
formula_36
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: