nLab spectrum of an operator

Idea

For a linear operator $A$ on a finite-dimensional complex vector space $V$, the spectrum of $A$ is simply the subset of the complex numbers consisting of the eigenvalues of $A$.

But in the case that $V$ is an infinite-dimensional complex separable Hilbert space, the (normal) eigenvalues only form the discrete spectrum. Instead, the full spectrum is the set of all $\lambda \in \mathbb{C}$ for which the resolvent $(A-\lambda I)^{-1}$ is not a bounded operator.

In other words, the spectrum is the complement of the subset of complex numbers for which the resolvent is a bounded operator.

The study of spectra of linear operators is quite rich, this is the topic of spectral theory.

Properties

If $A$ is a bounded linear operator on a complex separable Hilbert space, then the spectrum $\sigma(A)$ is a compact subset of $\mathbb{C}$.

The set $\sigma_d(A)$ of ordinary normal eigenvalues of $A$ is a subset of $\sigma(A)$ called the discrete spectrum of $A$. In particular case when $A$ is a compact operator the spectrum consists of the discrete spectrum only, except for possible addition of point $0$.

Related concepts

• discrete spectrum, essential spectrum

• Calkin algebra

• spectral gap

• spectral flow

• spectral theory

• min-entropy

References

• Vladimir Müller: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Operator Theory: Advances and Applications 139, Birkhäuser (2003, 2007) [doi:10.1007/978-3-7643-8265-0]

• Valter Moretti, Spectral Theory and Quantum Mechanics, Springer (2017) [doi:10.1007/978-3-319-70706-8]

  (with an eye towards applications in quantum mechanics)

See also:

• Wikipedia, Spectrum (functional_analysis)