Spectrum (functional analysis)

In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. 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, 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 the operator R − 0 (i.e. R itself) is not invertible: it is not surjective since any vector with non-zero first component is not in its range. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum. The notion of spectrum extends to densely defined unbounded operators. In this case a complex number λ is said to be in the spectrum of such an operator T:D→X (where D is dense in X) if there is no bounded inverse (λI − T)−1:X→D. If T is a closed operator (which includes the case that T is a bounded operator), boundedness of such inverses follows automatically if the inverse exists at all. 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. Let T {displaystyle T} be a bounded linear operator acting on a Banach space X {displaystyle X} over the complex scalar field C {displaystyle mathbb {C} } , and I {displaystyle I} be the identity operator on X {displaystyle X} . The spectrum of T {displaystyle T} is the set of all λ ∈ C {displaystyle lambda in mathbb {C} } for which the operator λ I − T {displaystyle lambda I-T} does not have an inverse that is a bounded linear operator. Since λ I − T {displaystyle lambda I-T} is a linear operator, the inverse is linear if it exists; and, by the bounded inverse theorem, it is bounded. Therefore, the spectrum consists precisely of those scalars λ {displaystyle lambda } for which λ I − T {displaystyle lambda I-T} is not bijective. The spectrum of a given operator T {displaystyle T} is often denoted σ ( T ) {displaystyle sigma (T)} , and its complement, the resolvent set, is denoted ρ ( T ) = C ∖ σ ( T ) {displaystyle ho (T)=mathbb {C} setminus sigma (T)} . ( ρ ( T ) {displaystyle ho (T)} is sometimes used to denote the spectral radius of T {displaystyle T} ) If λ {displaystyle lambda } is an eigenvalue of T {displaystyle T} , then the operator T − λ I {displaystyle T-lambda I} is not one-to-one, and therefore its inverse ( T − λ I ) − 1 {displaystyle (T-lambda I)^{-1}} is not defined. However, the inverse statement is not true: the operator T − λ I {displaystyle T-lambda I} may not have an inverse, even if λ {displaystyle lambda } is not an eigenvalue. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.