Spectrum (functional analysis)

In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number ${\displaystyle \lambda }$ is said to be in the spectrum of a bounded linear operator ${\displaystyle T}$ if ${\displaystyle T-\lambda I}$

• either has no set-theoretic inverse;
• or the set-theoretic inverse is either unbounded or defined on a non-dense subset.^[1]

Here, ${\displaystyle I}$ is the identity operator.

By the closed graph theorem, ${\displaystyle \lambda }$ is in the spectrum if and only if the bounded operator ${\displaystyle T-\lambda I:V\to V}$ is non-bijective on ${\displaystyle V}$.

The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.

The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator R on the Hilbert space ℓ²,

${\displaystyle (x_{1},x_{2},\dots )\mapsto (0,x_{1},x_{2},\dots ).}$

This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x₁=0, x₂=0, etc. On the other hand, 0 is in the spectrum because although the operator R − 0 (i.e. R itself) is invertible, the inverse is defined on a set which is not dense in ℓ². In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.

The notion of spectrum extends to unbounded (i.e. not necessarily bounded) operators. A complex number λ is said to be in the spectrum of an unbounded operator ${\displaystyle T:\,X\to X}$ defined on domain ${\displaystyle D(T)\subseteq X}$ if there is no bounded inverse ${\displaystyle (T-\lambda I)^{-1}:\,X\to D(T)}$ defined on the whole of ${\displaystyle X.}$ If T is closed (which includes the case when T is bounded), boundedness of ${\displaystyle (T-\lambda I)^{-1}}$ follows automatically from its existence.

The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.