Compact operator on Hilbert space

In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finiterank 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.
Contents
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 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 to Y (with the norm topology). (See (Zhu 2007, Theorem 1.14, p.11), and note in this reference that the uniform boundedness will apply in the situation where FX 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 normclosed, twosided, *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 S_{n} → S in the strong operator topology and T is compact, then S_{n}T converges to ST in norm. For example, consider the Hilbert space l^{2}(N), with standard basis {e_{n}}. Let P_{m} be the orthogonal projection on the linear span of {e_{1}...e_{m}}. The sequence {P_{m}} converges to the identity operator I strongly but not uniformly. Define T by Te_{n} = (1/n)^{2} · e_{n}. T is compact, and, as claimed above, P_{m}T → I T = T in the uniform operator topology: for all x,
Notice each P_{m} is a finiterank operator. Similar reasoning shows that if T is compact, then T is the uniform limit of some sequence of finiterank operators.
By the normclosedness 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.
Compact self adjoint operator
A bounded operator T on a Hilbert space H is said to be selfadjoint 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 selfadjoint 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 (ker T)^{⊥} of the kernel (which is equal to the closure of the range of T, for any bounded selfadjoint operator). These basic facts play an important role in the proof of the spectral theorem below.
The classification result for Hermitian n × n 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.
Spectral theorem
Theorem For every compact selfadjoint operator T on a real or complex Hilbert space H, there exists an orthonormal basis 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 {e_{n}} of eigenvectors of T, with corresponding eigenvalues {λ_{n}} ⊂ R, such that λ_{n} → 0.
In other words, a compact selfadjoint operator can be unitarily diagonalized. This is the spectral theorem.
When H is separable, one can mix the basis {e_{n}} with a countable orthonormal basis for the kernel of T, and obtain an orthonormal basis {f_{n}} for H, consisting of eigenvectors of T with real eigenvalues {μ_{n}} such that μ_{n} → 0.
Corollary For every compact selfadjoint operator T on a real or complex separable infinite dimensional Hilbert space H, there exists a countably infinite orthonormal basis {f_{n}} of H consisting of eigenvectors of T, with corresponding eigenvalues {μ_{n}} ⊂ R, such that μ_{n} → 0.
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 ƒ: R^{2n} → R 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 finitedimensional argument above, then apply induction. We first sketch the argument for matrices.
Since the closed unit sphere S in R^{2n} 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, Ty = λy.
However, the Lagrange multipliers do not generalize easily to the infinite dimensional case. Alternatively, let z ∈ C^{n} 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 h : R → R, h(t) = g(y + t z). By calculus, h ′(0) = 0, i.e.,
Let m = <Ty, y> / <y, y>. After some algebra the above expression becomes (Re denotes the real part of a complex number)
But z is arbitrary, therefore Ty − my = 0. This is the crux of proof for spectral theorem in the matricial case.
Details
Claim If T is a compact selfadjoint operator on a nonzero Hilbert space H and
then m(T) or −m(T) is an eigenvalue of T.
If m(T) = 0, then T = 0 by the polarization identity, and this case is clear. Consider the function ƒ : H → R 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 m(T) > 0. 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 λ = <λy, y> = <Ty, y> = ƒ(y) = m(T).
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 y ∈ B. 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 {y_{n}}. Then y_{n} converges to 0 weakly, but lim ƒ(y_{n}) = 1 ≠ 0 = ƒ(0).
Let T be a compact operator on a Hilbert space H. A finite (possibly empty) or countably infinite orthonormal sequence {e_{n}} of eigenvectors of T, with corresponding nonzero eigenvalues, is constructed by induction as follows. Let H_{0} = H and T_{0} = T. If m(T_{0}) = 0, then T = 0 and the construction stops without producing any eigenvector e_{n}. Suppose that orthonormal eigenvectors e_{0}, …, e_{n−1} of T have been found. Then E_{n} := span(e_{0}, …, e_{n−1}) is invariant under T, and by selfadjointness, the orthogonal complement H_{n} of E_{n} is an invariant subspace of T. Let T_{n} denote the restriction of T to H_{n}. If m(T_{n}) = 0, then T_{n} = 0, and the construction stops. Otherwise, by the claim applied to T_{n}, there is a norm one eigenvector e_{n} of T in H_{n}, with corresponding nonzero eigenvalue λ_{n} = ± m(T_{n}).
Let F = (span{e_{n}})^{⊥}, where {e_{n}} is the finite or infinite sequence constructed by the inductive process; by selfadjointness, 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 e_{m−1}, then F = H_{m} and S = T_{m} = 0 by construction. In the infinite case, compactness of T and the weakconvergence of e_{n} to 0 imply that T e_{n} = λ_{n} e_{n} → 0, therefore λ_{n} → 0. Since F is contained in H_{n} for every n, it follows that m(S) ≤ m(T_{n}) = λ_{n} for every n, hence m(S) = 0. This implies again that S = 0.
The fact that S = 0 means that F is contained in the kernel of T. Conversely, if x ∈ ker(T), then by selfadjointness, x is orthogonal to every eigenvector e_{n} with nonzero eigenvalue. It follows that F = ker(T), and that {e_{n}} 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 nontrivial 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 {λ_{n}} 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. 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 Φ : C(σ(T)) → L(H) such that Φ(1) = I and, if ƒ is the identity function ƒ(λ) = λ, then Φ(ƒ) = T. Moreover, σ(ƒ(T)) = ƒ(σ(T)).
The functional calculus map Φ is defined in a natural way: let {e_{n}} be an orthonormal basis of eigenvectors for H, with corresponding eigenvalues {λ_{n}}; for ƒ ∈ C(σ(T)), the operator Φ(ƒ), diagonal with respect to the orthonormal basis {e_{n}}, 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 Ψ = Φ. This shows that Φ is unique.
The more general continuous functional calculus can be defined for any selfadjoint (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 selfadjoint 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 nonzero invariant subspace 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) nonzero invariant subspace (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 .
(Lem)
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 nontrivial 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.
(Thm 1)
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 T_{0} is symmetric, we have that is a (countable) orthonormal set. It is also, by the decomposition we first stated, a basis for .
(Thm 2)
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 nontrivial 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.
(Thm 3)
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 uniqueuptoscale Haar measure on ). Consider the continuous shift action where . Then if were compact then there is a unique decomposition of H 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 1dimensional 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 (in particular, a subnormal operator) is normal.
Unitary operator
The spectrum of a unitary operator 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 U = I + C where C is compact. The equations U U* = U*U = I and C = U − I show that C is normal. The spectrum of C contains 0, and possibly, a finite set or a sequence tending to 0. Since U = I + C, the spectrum of U is obtained by shifting the spectrum of C by 1.
Examples
 Let H = L^{2}([0, 1]). The multiplication operator M defined by
is a bounded selfadjoint 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 T_{K} on H by
Then T_{K} is compact on H; it is a Hilbert–Schmidt operator.
 Suppose that the kernel K(x, y) satisfies the Hermiticity condition
Then T_{K} is compact and selfadjoint on H; if {φ_{n}} is an orthonormal basis of eigenvectors, with eigenvalues {λ_{n}}, it can be proved that
where the sum of the series of functions is understood as L^{2} convergence for the Lebesgue measure on [0, 1]^{2}. Mercer's theorem gives conditions under which the series converges to K(x, y) pointwise, and uniformly on [0, 1]^{2}.
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). If the compactness assumption is removed, operators need not have countable spectrum in general.
References
 J. Blank, P. Exner, and M. Havlicek, Hilbert Space Operators in Quantum Physics, American Institute of Physics, 1994.
 M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972.
 Zhu, Kehe (2007), Operator Theory in Function Spaces, Mathematical surveys and monographs, Vol. 138, American Mathematical Society, ISBN 9780821839652
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