The spectrum of a linear operator ${\displaystyle T}$ that operates on a Banach space ${\displaystyle X}$ is a fundamental concept of functional analysis. The spectrum consists of all scalars ${\displaystyle \lambda }$ such that the operator ${\displaystyle T-\lambda }$ does not have a bounded inverse on ${\displaystyle X}$. The spectrum has a standard decomposition into three parts:

This decomposition is relevant to the study of differential equations, and has applications to many branches of science and engineering. A well-known example from quantum mechanics is the explanation for the discrete spectral lines and the continuous band in the light emitted by excited atoms of hydrogen.

Let X be a Banach space, B(X) the family of bounded operators on X, and T ∈ B(X). By definition, a complex number λ is in the spectrum of T, denoted σ(T), if T − λ does not have an inverse in B(X).

If T − λ is one-to-one and onto, i.e. bijective, then its inverse is bounded; this follows directly from the open mapping theorem of functional analysis. So, λ is in the spectrum of T if and only if T − λ is not one-to-one or not onto. One distinguishes three separate cases:

So σ(T) is the disjoint union of these three sets, $${\displaystyle \sigma (T)=\sigma _{p}(T)\cup \sigma _{c}(T)\cup \sigma _{r}(T).}$$The complement of the spectrum ${\displaystyle \sigma (T)}$ is known as resolvent set ${\displaystyle \rho (T)}$ that is ${\displaystyle \rho (T)=\mathbb {C} \setminus \sigma (T)}$.

In addition, when T − λ does not have dense range, whether is injective or not, then λ is said to be in the compression spectrum of T, σ_cp(T). The compression spectrum consists of the whole residual spectrum and part of point spectrum.

The spectrum of an unbounded operator can be divided into three parts in the same way as in the bounded case, but because the operator is not defined everywhere, the definitions of domain, inverse, etc. are more involved.

Given a σ-finite measure space (S, Σ, μ), consider the Banach space L^p(μ). A function h: S → C is called essentially bounded if h is bounded μ-almost everywhere. An essentially bounded h induces a bounded multiplication operator T_h on L^p(μ): $${\displaystyle (T_{h}f)(s)=h(s)\cdot f(s).}$$