Compact operator on Hilbert space
Encyclopedia
In functional analysis
, compact operator
s on Hilbert space
s are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank operators in the uniform operator topology
. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. In contrast, the study of general operators on infinite dimensional spaces often requires a genuinely different approach.
For example, the spectral theory of compact operators
on Banach spaces takes a form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is normal
. A corresponding result holds for normal compact operators on Hilbert spaces. (More generally, the compactness assumption can be dropped. But, as stated above, the techniques used are less routine.)
This article will discuss a few results for compact operators on Hilbert space, starting with general properties before considering subclasses of compact operators.
. We list some general properties of compact operators.
If X and Y are Hilbert spaces (in fact X Banach and Y normed will suffice), then T:X → Y is compact if and only if it is continuous when viewed as a map from X with the weak topology
to Y (with the norm topology). (See , and note in this reference that the uniform boundedness will apply in the situation where F
X satisfies (
φ є Hom(X,K)) sup{x**(φ) = φ(x):x} < ∞, where K is the underlying field. The uniform boundedness principle applies since Hom(X,K) with the norm topology will be a Banach space, and the maps x**:Hom(X,K) → K are continuous homomorphisms with respect to this topology.)
The family of compact operators is a norm-closed, two-sided, *-ideal in L(H). Consequently, a compact operator T cannot have a bounded inverse if H is infinite dimensional. If ST = TS = I, then the identity operator would be compact, a contradiction.
If a sequence of bounded operators Sn → S in the strong operator topology
and T is compact, then SnT converges to ST in norm. For example, consider the Hilbert space l2(N), with standard basis {en}. Let Pm be the orthogonal projection on the linear span of {e1...em}. The sequence {Pm} converges to the identity operator I strongly but not uniformly. Define T by Ten = (1/n)2 · en. T is compact, and, as claimed above, PmT → I T = T in the uniform operator topology: for all x,
Notice each Pm is a finite-rank operator. Similar reasoning shows that if T is compact, then T is the uniform limit of some sequence of finite-rank operators.
By the norm-closedness of the ideal of compact operators, the converse is also true.
The quotient C*-algebra of L(H) modulo the compact operators is called the Calkin algebra
, in which one can consider properties of an operator up to compact perturbation.
if T = T∗, or equivalently,
It follows that <Tx, x> is real for every x ∈ H, thus eigenvalues of T, when they exist, are real. When a closed linear subspace L of H is invariant under T, then the restriction of T to L is a self-adjoint operator on L, and furthermore, the orthogonal complement L⊥ of L is also invariant under T. For example, the space H can be decomposed as orthogonal direct sum of two T–invariant closed linear subspaces: the kernel of T, and the orthogonal complement of the kernel (which is equal to the closure of the range of T, for any bounded self-adjoint operator). These basic facts play an important role in the proof of the spectral theorem below.
The classification result for Hermitian matrices is the spectral theorem
: If M = M*, then M is unitarily diagonalizable and the diagonalization of M has real entries. Let T be a compact self adjoint operator on a Hilbert space H. We will prove the same statement for T: the operator T can be diagonalized by an orthonormal set of eigenvectors, each of which corresponds to a real eigenvalue.
of H consisting of eigenvectors of T. More specifically, the orthogonal complement of the kernel of T admits, either a finite orthonormal basis of eigenvectors of T, or a countably infinite
orthonormal basis {en} of eigenvectors of T, with corresponding eigenvalues , such that .
In other words, a compact self-adjoint operator can be unitarily diagonalized. This is the spectral theorem.
When H is separable, one can mix the basis {en} with a countable
orthonormal basis for the kernel of T, and obtain an orthonormal basis {fn} for H, consisting of eigenvectors of T with real eigenvalues {μn} such that .
Corollary For every compact self-adjoint operator T on a real or complex separable infinite dimensional Hilbert space H, there exists a countably infinite orthonormal basis {fn} of H consisting of eigenvectors of T, with corresponding eigenvalues , such that .
Note In the finite dimensional case, part of the first approach works in much greater generailty; any square matrix, not necessarily Hermitian, has an eigenvector. This is
simply not true for general operators on a Hilbert spaces.
The spectral theorem for the compact self adjoint case can be obtained analogously: one finds an eigenvector by extending the second finite-dimensional argument above, then apply induction. We first sketch the argument for matrices.
Since the closed unit sphere S in R2n is compact, and ƒ is continuous, ƒ(S) is compact on the real line, therefore ƒ attains a maximum on S, at some unit vector y. By Lagrange's multiplier
theorem, y satisfies
for some λ. By Hermiticity, .
However, the Lagrange multipliers do not generalize easily to the infinite dimensional case. Alternatively, let z ∈ Cn be any vector. Notice that if a unit vector y maximizes <Tx, x> on the unit sphere (or on the unit ball), it also maximizes the Rayleigh quotient:
Consider the function , . By calculus, , i.e.,
Let m = . After some algebra the above expression becomes (Re denotes the real part of a complex number)
But z is arbitrary, therefore . This is the crux of proof for spectral theorem in the matricial case.
then m(T) or −m(T) is an eigenvalue of T.
If , then T = 0 by the polarization identity
, and this case is clear. Consider the function defined by ƒ(x) = <Tx, x>. Replacing T by −T if necessary, one may assume that the supremum of ƒ on the closed unit ball B ⊂ H is equal to . If ƒ attains its maximum m(T) on B at some unit vector y, then, by the same argument used for matrices, y is an eigenvector of T, with corresponding eigenvalue = .
By the Banach–Alaoglu theorem and the reflexivity of H, the closed unit ball B is weakly compact. Also, the compactness of T means (see above) that T:X with the weak topology → X with the norm topology, is continuous. These two facts imply that ƒ is continuous on B equipped with the weak topology, and ƒ attains therefore its maximum m on B at some . By maximality, ||y|| = 1, which in turn implies that y also maximizes the Rayleigh quotient g(x) (see above). This shows that y is an eigenvector of T, and ends the proof of the claim.
Note The compactness of T is crucial. In general, ƒ need not be continuous for the weak topology on the unit ball B. For example, let T be the identity operator, which is not compact when H is infinite dimensional. Take any orthonormal sequence {yn}. Then yn converges to 0 weakly, but lim ƒ(yn) = 1 ≠ 0 = ƒ(0).
Let T be a compact operator on a Hilbert space H. A finite (possibly empty) or countably infinite orthonormal sequence {en} of eigenvectors of T, with corresponding
non-zero eigenvalues, is constructed by induction as follows. Let H0 = H and T0 = T. If m(T0) = 0, then T = 0 and the construction stops without producing any eigenvector en. Suppose that orthonormal eigenvectors of T have been found. Then is invariant under T, and by self-adjointness, the orthogonal complement Hn of En is an invariant subspace of T. Let Tn denote the restriction of T to Hn. If m(Tn) = 0, then Tn = 0, and the construction stops. Otherwise, by the claim applied to Tn, there is a norm one eigenvector en of T in Hn, with corresponding non-zero eigenvalue λn = .
Let F = (span{en})⊥, where {en} is the finite or infinite sequence constructed by the inductive process; by self-adjointness, F is invariant under T. Let S denote the restriction of T to F. If the process was stopped after finitely many steps, with a last vector em−1, then F = Hm and S = Tm = 0 by construction. In the infinite case, compactness of T and the weak-convergence of en to 0 imply that = , therefore . Since F is contained in Hn for every n, it follows that m(S) ≤ m(Tn) = |λn| for every n, hence m(S) = 0. This implies again that .
The fact that S = 0 means that F is contained in the kernel of T. Conversely, if x ∈ ker(T), then by self-adjointness, x is orthogonal to every eigenvector en with non-zero eigenvalue. It follows that , and that {en} is an orthonormal basis for the orthogonal complement of the kernel of T. One can complete the diagonalization of T by selecting an orthonormal basis of the kernel. This proves the spectral theorem.
A shorter but more abstract proof goes as follows: by Zorn's lemma
, select U to be a maximal subset of H with the following three properties: all elements of U are eigenvectors of T, they have norm one, and any two distinct elements of U are orthogonal. Let F be the orthogonal complement of the linear span of U. If F ≠ {0}, it is a non-trivial invariant subspace of T, and by the initial claim there must exist a norm one eigenvector y of T in F. But then U ∪ {y} contradicts the maximality of U. It follows that F = {0}, hence span(U) is dense in H. This shows that U is an orthonormal basis of H consisting of eigenvectors of T.
Any spectral theorem can be reformulated in terms of a functional calculus
. In the present context we have:
Theorem Let C(σ(T)) denote the C*-algebra of continuous functions on σ(T). There exists a unique isometric homomorphism L(H) such that Φ(1) = I and, if ƒ is the identity function ƒ(λ) = λ, then . Moreover, .
The functional calculus map Φ is defined in a natural way: let {en} be an orthonormal basis of eigenvectors for H, with corresponding eigenvalues {λn}; for , the operator Φ(ƒ), diagonal with respect to the orthonormal basis {en}, is defined by setting
for every n. Since Φ(ƒ) is diagonal with respect to an orthonormal basis, its norm is equal to the supremum of the modulus of diagonal coefficients,
The other properties of Φ can be readily verified. Conversely, any homomorphism Ψ satisfying the requirements of the theorem must coincide with Φ when ƒ is a polynomial. By the Weierstrass approximation theorem, polynomial functions are dense in C(σ(T)), and it follows that . This shows that Φ is unique.
The more general continuous functional calculus
can be defined for any self-adjoint (or even normal, in the complex case) bounded linear operator on a Hilbert space. The compact case, described here, is a particularly simple instance of this functional calculus.
(e.g. the finite dimensional 
), and a commuting set 
of self-adjoint operators. Then under suitable conditions, can be simultaneously (unitarily) diagonalised. Viz., there exists an orthonormal basis 
consisting of common eigenvectors for the operators — i.e.
Lemma: Suppose all the operators in
are compact. Then every closed non-zero 
-invariant sub-space 
has a common eigenvector for 
.
Proof (Lemma):
Case I: all the operators have each exactly one eigenvalue. Then take any
of unit length. This is a common eigen vector.
Case II: there is some operator
with at least 2 eigen values. Then let 
. By compactness of 
and since 
, we have 
is a finite dimensional (and therefore closed) non-zero 
-invariant sub-space (because the operators all commute with 
, we have for 
and 
, that 
). In particular we definitely have 
. Thus we could in principle argue by induction over dimension, yielding that 
has a common eigenvector for 
.
Theorem 1: If all the operators in
are compact then the operators can be simultaneously (unitarily) diagonalised.
Proof (Thm 1): Consider
an orthonormal set of common eigen vectors for 
partially ordered by inclusion. This clearly has the Zorn property. So taking 
a maximal member, if 
is a basis for the whole hilbert space 
, we are done. If this were not the case, then letting 
, it is easy to see that this would be an 
-invariant non-trivial closed subspace; and thus by the lemma above, therein would lie a common eigen vector for the operators (necessarily orthogonal to 
). But then there would then be a proper extension of 
within 
; a contradiction to its maximality.
Theorem 2: If there is an injective compact operator in
; then the operators can be simultaneously (unitarily) diagonalised.
Proof (Thm 2): Fix
compact injective. Then we have, by the spectral theory of compact symmetric operators on hilbert spaces: 
, where 
is a discrete, countable set, and all the eigen spaces are finite dimensional. Since 
a commuting set, we have all the eigen spaces are invariant. Since the operators restricted to the eigen spaces (which are finite dimensional) are automatically all compact, we can apply Theorem 1 to each of these, and find orthonormal bases 
for the 
. Since 
is symmetric, we have that 
is a (countable) orthonormal set. It is also, by the decomposition we first stated, a basis for 
.
Theorem 3: If
a finite dimensional hilbert space, and 
a commutative set of operators, each of which is diagonalisable; then the operators can be simultaneously diagonalised.
Proof (Thm 3): Case I: all operators have exactly one eigenvalue. Then any basis for
will do. Case II: Fix 
an operator with at least two eigenvalues, and let 
so that 
is a symmetric operator. Now let 
an eigenvalue of 
. Then it is easy to see that 
and 
are non-trivial 
-invariant subspaces. By induction over dimension we have that there are linearly independent bases 
for the subspaces, which demonstrate that the operators in 
can be simultaneously diagonalisable on the subspaces. Clearly then 
demonstrates that the operators in 
can be simultaneously diagonalised.
Notice we did not have to directly use the machinery of matrices at all in this proof. There are other versions which do.
We can strengthen the above to the case where all the operators merely commute with their adjoint; in this case we remove the term "orthogonal" from the diagonalisation. There are weaker results for operators arising from representations due to Weyl–Peter. Let
be a fixed locally compact hausdorff group, and 
(the space of square integrable measurable functions with respect to the unique-up-to-scale Haar measure on 
). Consider the continuous shift action 
where 
. Then if 
were compact then there is a unique decomposition of 
into a countable direct sum of finite dimensional, irreducible, invariant subspaces (this is essentially diagonalisation of the family of operators 
). If 
were not compact, but were abelien, then diagonalisation is not achieved, but we get a unique continuous decomposition of 
into 1-dimensional invariant subspaces.
Let T be compact and T*T = TT*. Apply the Cartesian decomposition to T: define
The self adjoint compact operators R and J are called the real and imaginary parts of T respectively. T is compact means T*, consequently R and J, are compact. Furthermore, the normality of T implies R and J commute. Therefore they can be simultaneously diagonalized, from which follows the claim.
A hyponormal compact operator
(in particular, a subnormal operator
) is normal.
U lies on the unit circle in the complex plane; it could be the entire unit circle. However, if U is identity plus a compact perturbation, U has only countable spectrum, containing 1 and possibly, a finite set or a sequence tending to 1 on the unit circle. More precisely, suppose where C is compact. The equations and show that C is normal. The spectrum of C contains 0, and possibly, a finite set or a sequence tending to 0. Since , the spectrum of U is obtained by shifting the spectrum of C by 1.
is a bounded self-adjoint operator on H that has no eigenvector and hence, by the spectral theorem, can not
be compact.
Then TK is compact on H; it is a Hilbert–Schmidt operator.
Then TK is compact and self-adjoint on H; if {φn} is an orthonormal basis of eigenvectors, with eigenvalues {λn}, it can be proved that
where the sum of the series of functions is understood as L2 convergence for the Lebesgue measure . Mercer's theorem
gives conditions under which the series converges to K(x, y) pointwise, and uniformly .
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, compact operator
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
s on Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank operators in the uniform operator topology
Operator topology
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B of bounded linear operators on a Hilbert space H.-Introduction:...
. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. In contrast, the study of general operators on infinite dimensional spaces often requires a genuinely different approach.
For example, the spectral theory of compact operators
Spectral theory of compact operators
In functional analysis, compact operators are linear operators that map bounded sets to precompact sets. The set of compact operators acting on a Hilbert space H is the closure of the set of finite rank operators in the uniform operator topology. In general, operators on infinite dimensional spaces...
on Banach spaces takes a form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is normal
Normal operator
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operatorN:H\to Hthat commutes with its hermitian adjoint N*: N\,N^*=N^*N....
. A corresponding result holds for normal compact operators on Hilbert spaces. (More generally, the compactness assumption can be dropped. But, as stated above, the techniques used are less routine.)
This article will discuss a few results for compact operators on Hilbert space, starting with general properties before considering subclasses of compact operators.
Some general properties
Let H be a Hilbert space, L(H) be the bounded operators on H. T ∈ L(H) is a compact operator if the image of each bounded set under T is relatively compactRelatively compact subspace
In mathematics, a relatively compact subspace Y of a topological space X is a subset whose closure is compact....
. We list some general properties of compact operators.
If X and Y are Hilbert spaces (in fact X Banach and Y normed will suffice), then T:X → Y is compact if and only if it is continuous when viewed as a map from X with the weak topology
Weak convergence (Hilbert space)
In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology.-Definition:A sequence of points in a Hilbert space H is said to converge weakly to a point x in H if...
to Y (with the norm topology). (See , and note in this reference that the uniform boundedness will apply in the situation where F
X satisfies (φ є Hom(X,K)) sup{x**(φ) = φ(x):x} < ∞, where K is the underlying field. The uniform boundedness principle applies since Hom(X,K) with the norm topology will be a Banach space, and the maps x**:Hom(X,K) → K are continuous homomorphisms with respect to this topology.)The family of compact operators is a norm-closed, two-sided, *-ideal in L(H). Consequently, a compact operator T cannot have a bounded inverse if H is infinite dimensional. If ST = TS = I, then the identity operator would be compact, a contradiction.
If a sequence of bounded operators Sn → S in the strong operator topology
Strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
and T is compact, then SnT converges to ST in norm. For example, consider the Hilbert space l2(N), with standard basis {en}. Let Pm be the orthogonal projection on the linear span of {e1...em}. The sequence {Pm} converges to the identity operator I strongly but not uniformly. Define T by Ten = (1/n)2 · en. T is compact, and, as claimed above, PmT → I T = T in the uniform operator topology: for all x,
Notice each Pm is a finite-rank operator. Similar reasoning shows that if T is compact, then T is the uniform limit of some sequence of finite-rank operators.
By the norm-closedness of the ideal of compact operators, the converse is also true.
The quotient C*-algebra of L(H) modulo the compact operators is called the Calkin algebra
Calkin algebra
In functional analysis, the Calkin algebra, named after John Wilson Calkin, is the quotient of B, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K of compact operators....
, in which one can consider properties of an operator up to compact perturbation.
Compact self adjoint operator
A bounded operator T on a Hilbert space H is said to be self-adjointSelf-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
if T = T∗, or equivalently,
It follows that <Tx, x> is real for every x ∈ H, thus eigenvalues of T, when they exist, are real. When a closed linear subspace L of H is invariant under T, then the restriction of T to L is a self-adjoint operator on L, and furthermore, the orthogonal complement L⊥ of L is also invariant under T. For example, the space H can be decomposed as orthogonal direct sum of two T–invariant closed linear subspaces: the kernel of T, and the orthogonal complement of the kernel (which is equal to the closure of the range of T, for any bounded self-adjoint operator). These basic facts play an important role in the proof of the spectral theorem below.
The classification result for Hermitian matrices is the spectral theorem
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
: If M = M*, then M is unitarily diagonalizable and the diagonalization of M has real entries. Let T be a compact self adjoint operator on a Hilbert space H. We will prove the same statement for T: the operator T can be diagonalized by an orthonormal set of eigenvectors, each of which corresponds to a real eigenvalue.
Spectral theorem
Theorem For every compact self-adjoint operator T on a real or complex Hilbert space H, there exists an orthonormal basisOrthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
of H consisting of eigenvectors of T. More specifically, the orthogonal complement of the kernel of T admits, either a finite orthonormal basis of eigenvectors of T, or a countably infinite
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
orthonormal basis {en} of eigenvectors of T, with corresponding eigenvalues , such that .
In other words, a compact self-adjoint operator can be unitarily diagonalized. This is the spectral theorem.
When H is separable, one can mix the basis {en} with a countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
orthonormal basis for the kernel of T, and obtain an orthonormal basis {fn} for H, consisting of eigenvectors of T with real eigenvalues {μn} such that .
Corollary For every compact self-adjoint operator T on a real or complex separable infinite dimensional Hilbert space H, there exists a countably infinite orthonormal basis {fn} of H consisting of eigenvectors of T, with corresponding eigenvalues , such that .
The idea
Proving the spectral theorem for a Hermitian n × n matrix T hinges on showing the existence of one eigenvector x. Once this is done, Hermiticity implies that both the linear span and orthogonal complement of x are invariant subspaces of T. The desired result is then obtained by iteration. The existence of an eigenvector can be shown in at least two ways:- One can argue algebraically: The characteristic polynomial of T has a complex root, therefore T has an eigenvalue with a corresponding eigenvector. Or,
- The eigenvalues can be characterized variationally: The largest eigenvalue is the maximum on the closed unit sphere of the function defined by ƒ(x) = x*Tx = <Tx, x>.
Note In the finite dimensional case, part of the first approach works in much greater generailty; any square matrix, not necessarily Hermitian, has an eigenvector. This is
simply not true for general operators on a Hilbert spaces.
The spectral theorem for the compact self adjoint case can be obtained analogously: one finds an eigenvector by extending the second finite-dimensional argument above, then apply induction. We first sketch the argument for matrices.
Since the closed unit sphere S in R2n is compact, and ƒ is continuous, ƒ(S) is compact on the real line, therefore ƒ attains a maximum on S, at some unit vector y. By Lagrange's multiplier
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
theorem, y satisfies
for some λ. By Hermiticity, .
However, the Lagrange multipliers do not generalize easily to the infinite dimensional case. Alternatively, let z ∈ Cn be any vector. Notice that if a unit vector y maximizes <Tx, x> on the unit sphere (or on the unit ball), it also maximizes the Rayleigh quotient:
Consider the function , . By calculus, , i.e.,
Let m = . After some algebra the above expression becomes (Re denotes the real part of a complex number)
But z is arbitrary, therefore . This is the crux of proof for spectral theorem in the matricial case.
Details
Claim If T is a compact self-adjoint operator on a non-zero Hilbert space H andthen m(T) or −m(T) is an eigenvalue of T.
If , then T = 0 by the polarization identity
Polarization identity
In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Let \|x\| \, denote the norm of vector x and \langle x, \ y \rangle \, the inner product of vectors x and y...
, and this case is clear. Consider the function defined by ƒ(x) = <Tx, x>. Replacing T by −T if necessary, one may assume that the supremum of ƒ on the closed unit ball B ⊂ H is equal to . If ƒ attains its maximum m(T) on B at some unit vector y, then, by the same argument used for matrices, y is an eigenvector of T, with corresponding eigenvalue = .
By the Banach–Alaoglu theorem and the reflexivity of H, the closed unit ball B is weakly compact. Also, the compactness of T means (see above) that T:X with the weak topology → X with the norm topology, is continuous. These two facts imply that ƒ is continuous on B equipped with the weak topology, and ƒ attains therefore its maximum m on B at some . By maximality, ||y|| = 1, which in turn implies that y also maximizes the Rayleigh quotient g(x) (see above). This shows that y is an eigenvector of T, and ends the proof of the claim.
Note The compactness of T is crucial. In general, ƒ need not be continuous for the weak topology on the unit ball B. For example, let T be the identity operator, which is not compact when H is infinite dimensional. Take any orthonormal sequence {yn}. Then yn converges to 0 weakly, but lim ƒ(yn) = 1 ≠ 0 = ƒ(0).
Let T be a compact operator on a Hilbert space H. A finite (possibly empty) or countably infinite orthonormal sequence {en} of eigenvectors of T, with corresponding
non-zero eigenvalues, is constructed by induction as follows. Let H0 = H and T0 = T. If m(T0) = 0, then T = 0 and the construction stops without producing any eigenvector en. Suppose that orthonormal eigenvectors of T have been found. Then is invariant under T, and by self-adjointness, the orthogonal complement Hn of En is an invariant subspace of T. Let Tn denote the restriction of T to Hn. If m(Tn) = 0, then Tn = 0, and the construction stops. Otherwise, by the claim applied to Tn, there is a norm one eigenvector en of T in Hn, with corresponding non-zero eigenvalue λn = .
Let F = (span{en})⊥, where {en} is the finite or infinite sequence constructed by the inductive process; by self-adjointness, F is invariant under T. Let S denote the restriction of T to F. If the process was stopped after finitely many steps, with a last vector em−1, then F = Hm and S = Tm = 0 by construction. In the infinite case, compactness of T and the weak-convergence of en to 0 imply that = , therefore . Since F is contained in Hn for every n, it follows that m(S) ≤ m(Tn) = |λn| for every n, hence m(S) = 0. This implies again that .
The fact that S = 0 means that F is contained in the kernel of T. Conversely, if x ∈ ker(T), then by self-adjointness, x is orthogonal to every eigenvector en with non-zero eigenvalue. It follows that , and that {en} is an orthonormal basis for the orthogonal complement of the kernel of T. One can complete the diagonalization of T by selecting an orthonormal basis of the kernel. This proves the spectral theorem.
A shorter but more abstract proof goes as follows: by Zorn's lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
, select U to be a maximal subset of H with the following three properties: all elements of U are eigenvectors of T, they have norm one, and any two distinct elements of U are orthogonal. Let F be the orthogonal complement of the linear span of U. If F ≠ {0}, it is a non-trivial invariant subspace of T, and by the initial claim there must exist a norm one eigenvector y of T in F. But then U ∪ {y} contradicts the maximality of U. It follows that F = {0}, hence span(U) is dense in H. This shows that U is an orthonormal basis of H consisting of eigenvectors of T.
Functional calculus
If T is compact on an infinite dimensional Hilbert space H, then T is not invertible, hence σ(T), the spectrum of T, always contains 0. The spectral theorem shows that σ(T) consists of the eigenvalues {λn} of T, and of 0 (if 0 is not already an eigenvalue). The set σ(T) is a compact subset of the real line, and the eigenvalues are dense in σ(T).Any spectral theorem can be reformulated in terms of a functional calculus
Functional calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch of the field of functional analysis, connected with spectral theory. In mathematics, a functional calculus is a theory allowing one to apply mathematical...
. In the present context we have:
Theorem Let C(σ(T)) denote the C*-algebra of continuous functions on σ(T). There exists a unique isometric homomorphism L(H) such that Φ(1) = I and, if ƒ is the identity function ƒ(λ) = λ, then . Moreover, .
The functional calculus map Φ is defined in a natural way: let {en} be an orthonormal basis of eigenvectors for H, with corresponding eigenvalues {λn}; for , the operator Φ(ƒ), diagonal with respect to the orthonormal basis {en}, is defined by setting
for every n. Since Φ(ƒ) is diagonal with respect to an orthonormal basis, its norm is equal to the supremum of the modulus of diagonal coefficients,
The other properties of Φ can be readily verified. Conversely, any homomorphism Ψ satisfying the requirements of the theorem must coincide with Φ when ƒ is a polynomial. By the Weierstrass approximation theorem, polynomial functions are dense in C(σ(T)), and it follows that . This shows that Φ is unique.
The more general continuous functional calculus
Continuous functional calculus
In mathematics, the continuous functional calculus of operator theory and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. More precisely,Theorem...
can be defined for any self-adjoint (or even normal, in the complex case) bounded linear operator on a Hilbert space. The compact case, described here, is a particularly simple instance of this functional calculus.
Simultaneous diagonalisation
Consider an Hilbert space (e.g. the finite dimensional ), and a commuting set of self-adjoint operators. Then under suitable conditions, can be simultaneously (unitarily) diagonalised. Viz., there exists an orthonormal basis consisting of common eigenvectors for the operators — i.e.Lemma: Suppose all the operators in
are compact. Then every closed non-zero -invariant sub-space has a common eigenvector for .Proof (Lemma):
Case I: all the operators have each exactly one eigenvalue. Then take any
of unit length. This is a common eigen vector.Case II: there is some operator
with at least 2 eigen values. Then let . By compactness of and since , we have is a finite dimensional (and therefore closed) non-zero -invariant sub-space (because the operators all commute with , we have for and , that ). In particular we definitely have . Thus we could in principle argue by induction over dimension, yielding that has a common eigenvector for .Theorem 1: If all the operators in
are compact then the operators can be simultaneously (unitarily) diagonalised.Proof (Thm 1): Consider
an orthonormal set of common eigen vectors for partially ordered by inclusion. This clearly has the Zorn property. So taking a maximal member, if is a basis for the whole hilbert space , we are done. If this were not the case, then letting , it is easy to see that this would be an -invariant non-trivial closed subspace; and thus by the lemma above, therein would lie a common eigen vector for the operators (necessarily orthogonal to ). But then there would then be a proper extension of within ; a contradiction to its maximality.Theorem 2: If there is an injective compact operator in
; then the operators can be simultaneously (unitarily) diagonalised.Proof (Thm 2): Fix
compact injective. Then we have, by the spectral theory of compact symmetric operators on hilbert spaces: , where is a discrete, countable set, and all the eigen spaces are finite dimensional. Since a commuting set, we have all the eigen spaces are invariant. Since the operators restricted to the eigen spaces (which are finite dimensional) are automatically all compact, we can apply Theorem 1 to each of these, and find orthonormal bases for the . Since is symmetric, we have that is a (countable) orthonormal set. It is also, by the decomposition we first stated, a basis for .Theorem 3: If
a finite dimensional hilbert space, and a commutative set of operators, each of which is diagonalisable; then the operators can be simultaneously diagonalised.Proof (Thm 3): Case I: all operators have exactly one eigenvalue. Then any basis for
will do. Case II: Fix an operator with at least two eigenvalues, and let so that is a symmetric operator. Now let an eigenvalue of . Then it is easy to see that and are non-trivial -invariant subspaces. By induction over dimension we have that there are linearly independent bases for the subspaces, which demonstrate that the operators in can be simultaneously diagonalisable on the subspaces. Clearly then demonstrates that the operators in can be simultaneously diagonalised.Notice we did not have to directly use the machinery of matrices at all in this proof. There are other versions which do.
We can strengthen the above to the case where all the operators merely commute with their adjoint; in this case we remove the term "orthogonal" from the diagonalisation. There are weaker results for operators arising from representations due to Weyl–Peter. Let
be a fixed locally compact hausdorff group, and (the space of square integrable measurable functions with respect to the unique-up-to-scale Haar measure on ). Consider the continuous shift action where . Then if were compact then there is a unique decomposition of into a countable direct sum of finite dimensional, irreducible, invariant subspaces (this is essentially diagonalisation of the family of operators ). If were not compact, but were abelien, then diagonalisation is not achieved, but we get a unique continuous decomposition of into 1-dimensional invariant subspaces.Compact normal operator
The family of Hermitian matrices is a proper subset of matrices that are unitarily diagonalizable. A matrix M is unitarily diagonalizable if and only if it is normal, i.e. M*M = MM*. Similar statements hold for compact normal operators.Let T be compact and T*T = TT*. Apply the Cartesian decomposition to T: define
The self adjoint compact operators R and J are called the real and imaginary parts of T respectively. T is compact means T*, consequently R and J, are compact. Furthermore, the normality of T implies R and J commute. Therefore they can be simultaneously diagonalized, from which follows the claim.
A hyponormal compact operator
Hyponormal operator
In mathematics, especially operator theory, a hyponormal operator is a generalization of a normal operator. In general, a bounded linear operator T on a complex Hilbert space H is said to be p-hyponormal In mathematics, especially operator theory, a hyponormal operator is a generalization of a...
(in particular, a subnormal operator
Subnormal operator
In mathematics, especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples of subnormal operators are isometries and Toeplitz operators with analytic symbols.-Definition:Let H be a Hilbert space...
) is normal.
Unitary operator
The spectrum of a unitary operatorUnitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
U lies on the unit circle in the complex plane; it could be the entire unit circle. However, if U is identity plus a compact perturbation, U has only countable spectrum, containing 1 and possibly, a finite set or a sequence tending to 1 on the unit circle. More precisely, suppose where C is compact. The equations and show that C is normal. The spectrum of C contains 0, and possibly, a finite set or a sequence tending to 0. Since , the spectrum of U is obtained by shifting the spectrum of C by 1.
Examples
- Let H = L2([0, 1])Lp spaceIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
. The multiplication operator M defined by
is a bounded self-adjoint operator on H that has no eigenvector and hence, by the spectral theorem, can not
be compact.
- Let K(x, y) be square integrable on [0, 1]2 and define TK on H by
Then TK is compact on H; it is a Hilbert–Schmidt operator.
- Suppose that the kernel K(x, y) satisfies the Hermiticity condition
Then TK is compact and self-adjoint on H; if {φn} is an orthonormal basis of eigenvectors, with eigenvalues {λn}, it can be proved that
where the sum of the series of functions is understood as L2 convergence for the Lebesgue measure . Mercer's theorem
Mercer's theorem
In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in , is one of the most notable results of the work of James Mercer...
gives conditions under which the series converges to K(x, y) pointwise, and uniformly .
See also
- Singular value decomposition#Bounded operators on Hilbert spaces. The notion of singular values can be extended from matrices to compact operators.
- Decomposition of spectrum (functional analysis)Decomposition of spectrum (functional analysis)In mathematics, especially functional analysis, the spectrum of an operator generalizes the notion of eigenvalues. Given an operator, it is sometimes useful to break up the spectrum into various parts...
. If the compactness assumption is removed, operators need not have countable spectrum in general.
- Calkin algebraCalkin algebraIn functional analysis, the Calkin algebra, named after John Wilson Calkin, is the quotient of B, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K of compact operators....