Define spectrum and resolvent set of a bounded operator. Give an example of each.
Answer (1)
In mathematics, particularly in functional analysis, the concepts of the spectrum and resolvent set are important when studying linear operators on a Banach space or a Hilbert space.
Spectrum of a Bounded Operator
The spectrum of a bounded linear operator T on a Banach space X is the set of all complex numbers λ for which the operator T − λ I is not invertible, where I is the identity operator on X .
The spectrum σ ( T ) can be divided into three parts:
Point Spectrum (Eigenvalues) : These are values of λ for which T − λ I is not one-to-one.
Continuous Spectrum : These are values of λ for which T − λ I is one-to-one and onto, but the inverse is not bounded.
Residual Spectrum : These are values of λ for which T − λ I is one-to-one, and the inverse is everywhere defined but not bounded.
Example of Spectrum:
Consider the operator T : C n → C n defined by a matrix A . The spectrum σ ( T ) consists of the eigenvalues of A .
Resolvent Set of a Bounded Operator
The resolvent set ρ ( T ) of an operator T is the set of all complex numbers λ for which T − λ I is invertible.
The resolvent set is precisely the complement of the spectrum in the complex plane: ρ ( T ) = C ∖ σ ( T ) .
Example of Resolvent Set:
Continuing with the matrix A example, if λ is not an eigenvalue of A , then λ ∈ ρ ( T ) , and ( A − λ I ) − 1 exists.
By exploring these definitions, students can gain a deeper understanding of operator theory, which has applications in various mathematical and physical theories.
