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 λ is said to be in the spectrum of a bounded linear operator "T" if formula_1 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 "x"1=0, "x"2=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 unbounded operators. In this case a complex number "λ" is said to be in the spectrum of an operator formula_3 defined on domain formula_4 if there is no bounded inverse formula_5. 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.
Spectrum of a bounded operator.
Definition.
Let formula_6 be a bounded linear operator acting on a Banach space formula_7 over the complex scalar field formula_8, and formula_9 be the identity operator on formula_7. The spectrum of formula_6 is the set of all formula_12 for which the operator formula_1 does not have an inverse that is a bounded linear operator.
Since formula_1 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 formula_15 for which formula_1 is not bijective.
The spectrum of a given operator formula_6 is often denoted formula_18, and its complement, the resolvent set, is denoted formula_19. (formula_20 is sometimes used to denote the spectral radius of formula_6)
Relation to eigenvalues.
If formula_15 is an eigenvalue of formula_6, then the operator formula_1 is not one-to-one, and therefore its inverse formula_25 is not defined. However, the inverse statement is not true: the operator formula_26 may not have an inverse, even if formula_15 is not an eigenvalue. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.
For example, consider the Hilbert space formula_28, that consists of all bi-infinite sequences of real numbers
that have a finite sum of squares formula_30. The bilateral shift operator formula_6 simply displaces every element of the sequence by one position; namely if formula_32 then formula_33 for every integer formula_34. The eigenvalue equation formula_35 has no solution in this space, since it implies that all the values formula_36 have the same absolute value (if formula_37) or are a geometric progression (if formula_38); either way, the sum of their squares would not be finite. However, the operator formula_1 is not invertible if formula_40. For example, the sequence formula_41 such that formula_42 is in formula_28; but there is no sequence formula_44 in formula_28 such that formula_46 (that is, formula_47 for all formula_34).
Basic properties.
The spectrum of a bounded operator "T" is always a closed, bounded and non-empty subset of the complex plane.
If the spectrum were empty, then the "resolvent function"
would be defined everywhere on the complex plane and bounded. But it can be shown that the resolvent function "R" is holomorphic on its domain. By the vector-valued version of Liouville's theorem, this function is constant, thus everywhere zero as it is zero at infinity. This would be a contradiction.
The boundedness of the spectrum follows from the Neumann series expansion in "λ"; the spectrum "σ"("T") is bounded by ||"T"||. A similar result shows the closedness of the spectrum.
The bound ||"T"|| on the spectrum can be refined somewhat. The "spectral radius", "r"("T"), of "T" is the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum "σ"("T") inside of it, i.e.
Spectrum of the adjoint operator.
Let "X" be a Banach space and formula_3 a closed linear operator with dense domain formula_4.
If "X*" is the dual space of "X", and formula_59 is the hermitian adjoint of "T", then
Theorem For a bounded (or, more generally, closed and densely defined) operator "T", formula_61.
(T)</math>. So formula_62 is not dense in "X". 
By the Hahn–Banach theorem, there exists a non-zero formula_63 that vanishes on formula_62. 
For all "x" ∈ "X",
Therefore, formula_66 and formula_67 is an eigenvalue of "T*". 
This shows the former inclusion.
Next suppose that formula_68 with formula_63, formula_70, i.e.
If formula_72 is dense in "X", then "φ" must be the zero functional, a contradiction. 
The claim is proved. 
We also get formula_73 by the following argument: "X" embeds isometrically into "X**". 
Therefore, for every non-zero element in the kernel of formula_1 there exists a non-zero element in "X**" which vanishes on formula_75. 
Thus formula_76 can not be dense.
Furthermore, if "X" is reflexive, we have formula_77.
Spectra of particular classes of operators.
Compact operators.
If "T" is a compact operator, or, more generally, an inessential operator, then it can be shown that the spectrum is countable, that zero is the only possible accumulation point, and that any nonzero "λ" in the spectrum is an eigenvalue.
Quasinilpotent operators.
A bounded operator formula_78 is quasinilpotent if formula_79 as formula_80 (in other words, if the spectral radius of "A" equals zero). Such operators could equivalently be characterized by the condition
An example of such an operator is formula_82, formula_83 for formula_84.
Self-adjoint operators.
If "X" is a Hilbert space and "T" is a self-adjoint operator (or, more generally, a normal operator), then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).
For self-adjoint operators, one can use spectral measures to define a decomposition of the spectrum into absolutely continuous, pure point, and singular parts.
Spectrum of a real operator.
The definitions of the resolvent and spectrum can be extended to any continuous linear operator formula_6 acting on a Banach space formula_7 over the real field formula_87 (instead of the complex field formula_8) via its complexification formula_89. In this case we define the resolvent set formula_20 as the set of all formula_91 such that formula_92 is invertible as an operator acting on the complexified space formula_93; then we define formula_94.
Real spectrum.
The "real spectrum" of a continuous linear operator formula_6 acting on a real Banach space formula_7, denoted formula_97, is defined as the set of all formula_98 for which formula_1 fails to be invertible in the real algebra of bounded linear operators acting on formula_7. In this case we have formula_101. Note that the real spectrum may or may not coincide with the complex spectrum. In particular, the real spectrum could be empty.
Spectrum of a unital Banach algebra.
Let "B" be a complex Banach algebra containing a unit "e". Then we define the spectrum "σ"("x") (or more explicitly "σ""B"("x")) of an element "x" of "B" to be the set of those complex numbers "λ" for which "λe" − "x" is not invertible in "B". This extends the definition for bounded linear operators "B"("X") on a Banach space "X", since "B"("X") is a Banach algebra.