Compact Operators

Introduction

Among the bounded operators of a Hilbert space, the compact operators are the ones that come as close as an infinite-dimensional operator can to having finite rank: they send bounded sets to relatively compact sets, and on a Hilbert space they are exactly the norm limits of the finite-rank operators. The compact operators are more than a class: they form a closed two-sided self-adjoint ideal of $B(H)$, and it is the only proper closed ideal, so the quotient $B(H)/K(H)$ is the simple Calkin algebra. Their spectral theory is the discrete theory that the finite-dimensional spectral theorem extends to without loss, and the rate at which the singular values decay organises them into the Schatten classes $S_p$, of which the trace class $S_1$ and the Hilbert–Schmidt class $S_2$ are the two that carry a trace pairing.

This article assumes the algebra $B(H)$, the adjoint and the three topologies of Bounded Operators on a Hilbert Space; the compact self-adjoint spectral theorem and the Fredholm alternative are recalled from Banach and Hilbert Spaces, where they are proved, and are used here as the entry to the general Riesz–Schauder theory. The finite-rank operators are the elementary objects; the von Neumann algebra generated by a compact operator and the ideals of a general operator algebra belong to Operator Algebras (Part II), and the index of a Fredholm operator belongs to Fredholm Theory below, where the quotient algebra is developed. The trace-class duality with the weak operator topology is the predual statement used by Bounded Operators on a Hilbert Space.

Throughout, $H$ is a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, $\langle\cdot,\cdot\rangle$ is linear in the first argument, $B(H)$ is the bounded operators with operator norm $\|\cdot\|$, $F(H)$ the finite-rank operators, and $T^*$ is the adjoint. A compact operator is written $K$ or $T$; its singular values are $s_n(T)$, and the Schatten classes are $S_p(H)$.

Definition and Elementary Properties

Definition. An operator $T\in B(H)$ is compact if the image $T(B)$ of the closed unit ball $B=\{x:\|x\|\le1\}$ is relatively compact in $H$. The set of compact operators is written $K(H)$.

Proposition (sequential form). $T$ is compact exactly when every bounded sequence $(x_n)$ in $H$ has a subsequence $(x_{n_k})$ for which $T x_{n_k}$ converges.

Proof. A subset of a metric space is relatively compact exactly when every sequence in it has a convergent subsequence; applying this to $T(B)$ and rescaling by $\|x_n\|$ gives the equivalence.

Proposition (closure and limits of finite rank). $K(H)$ is a closed linear subspace of $B(H)$ containing $F(H)$, and

$$ K(H)=\overline{F(H)}^{\ \|\cdot\|} . $$

An operator is compact exactly when it is the norm limit of a sequence of finite-rank operators.

Proof. Linear combinations of compact operators are compact and a norm limit of compact operators is compact, so $K(H)$ is a closed subspace containing $F(H)$. For the density, let $T$ be compact and let $P_\alpha$ be the projection onto a finite-dimensional reducing subspace spanned by the vectors $x_1,\dots,x_n$ on which $\|Tx_i\|$ is nearly maximal; the standard approximation argument over a finite $\varepsilon$-net of $T(B)$ produces $P$ of finite rank with $\|T-TP\|<\varepsilon$, and $TP$ has finite rank.

Proposition (ideal property and involution). For $S\in B(H)$ and $T\in K(H)$,

$$ ST\in K(H),\qquad TS\in K(H),\qquad T^*\in K(H), $$

so $K(H)$ is a two-sided ideal of $B(H)$, self-adjoint under the involution.

Proof. $S$ and $T$ carry the bounded ball into a bounded set and then into a relatively compact set, giving $ST$; for $TS$ one uses that $S^*$ is bounded, $T^*T$ is compact and $TS=(S^*T^*)^*$. The adjoint is compact because $T$ is approximated by finite ranks, whose adjoints have finite rank.

Proposition (the identity is not compact). $I$ is compact exactly when $H$ is finite-dimensional. Consequently $K(H)=B(H)$ in finite dimension, and $K(H)$ is a proper ideal in infinite dimension.

Proof. The closed unit ball is compact exactly when $H$ is finite-dimensional; for the infinite-dimensional case the orthonormal sequence $e_n$ has no convergent subsequence in norm, so $I$ is not compact and $K(H)$ is proper by the ideal property applied to $K(H)=B(H)$.

The Ideal Structure and the Calkin Algebra

Theorem. In infinite dimension $K(H)$ is the unique maximal proper two-sided ideal, hence the unique proper closed two-sided ideal, of $B(H)$, and the quotient

$$ \mathcal{C}(H)=B(H)/K(H) $$

is a $C^*$-algebra, the Calkin algebra, with a faithful trace-free simple structure; it is simple and has no nontrivial ideals.

Proof. Every proper two-sided ideal of $B(H)$ is contained in $K(H)$: if an ideal contains an operator that is not compact, then a standard argument produces from it a rank-one projection, and the ideal generated by a rank-one projection is all of $B(H)$. Since $K(H)$ is closed and the quotient of a $C^*$-algebra by a closed ideal is a $C^*$-algebra, $\mathcal{C}(H)$ is a $C^*$-algebra, and the maximality of $K(H)$ makes it simple.

Remark. The Calkin algebra is the operator-theoretic home of the Fredholm index: an operator is Fredholm exactly when its class in $\mathcal{C}(H)$ is invertible, and the index is a homomorphism from the invertible group of $\mathcal{C}(H)$ to $\mathbb{Z}$. The index theory is Fredholm Theory below, and the $K$-theory of the Calkin algebra belongs to the index theorem of Part IV; neither is used here.

Proposition. The finite-rank operators are the minimal nonzero two-sided ideal, and every nonzero two-sided ideal of $B(H)$ contains $F(H)$ and is contained in $K(H)$.

Proof. A nonzero ideal contains a nonzero finite-rank operator, hence a rank-one projection, and the two-sided ideal generated by $F(H)$ is $F(H)$ itself; the containment $F(H)\subseteq J\subseteq K(H)$ follows from the minimality of the rank-one ideal and the first theorem.

The Spectral Theory of a Compact Operator

Theorem (Riesz–Schauder). Let $K\in K(H)$. Then:

(i) $\sigma(K)$ is a countable compact subset of $\mathbb{C}$ whose only possible accumulation point is $0$, and every nonzero $\lambda\in\sigma(K)$ is an eigenvalue of finite algebraic multiplicity;

(ii) the nonzero eigenspaces of $K$ are finite-dimensional and mutually orthogonal when $K$ is normal, and the eigenspaces at distinct nonzero eigenvalues are linearly independent with the root subspaces finite-dimensional;

(iii) $K-\lambda I$ is Fredholm for every $\lambda\neq0$, and the Fredholm alternative describes the solvability of $Kx-\lambda x=y$.

Proof. The compactness of $K$ makes $K(B)$ totally bounded, and an application of the spectral theorem to $K^*K$ gives the polar form $K=U|K|$ with $|K|$ compact and self-adjoint; the spectral theorem for a compact self-adjoint operator, quoted from Banach and Hilbert Spaces, gives the discrete spectrum of $|K|$, and the singular values tend to $0$. The Fredholm statement is the Fredholm alternative of the same article.

Corollary (the self-adjoint case). If $K$ is compact and self-adjoint then there is an orthonormal basis $(e_n)$ of $H$ with $K e_n=\lambda_n e_n$ and $\lambda_n\in\mathbb{R}$, $|\lambda_1|\ge|\lambda_2|\ge\cdots$, $\lambda_n\to0$ if the basis is infinite, and

$$ K=\sum_n\lambda_n\langle\cdot,e_n\rangle e_n $$

in operator norm.

Remark. The statement that a compact normal operator is diagonalised by the eigenvectors of its modulus is the reason the compact operators are said to have a discrete spectrum: the essential spectrum of a compact operator is $\{0\}$, and the continuous part of the spectrum is empty. The essential spectrum is the spectrum of the image in the Calkin algebra, and its computation is Fredholm Theory.

The Schatten Classes

Definition. For $T\in K(H)$ the singular values are the non-increasing sequence $s_1\ge s_2\ge\cdots\ge0$ of eigenvalues of $|T|=(T^*T)^{1/2}$, enumerated with multiplicity; $s_1=\|T\|$ and $s_n\downarrow0$. For $1\le p<\infty$ the Schatten $p$-class is

$$ S_p(H)=\Bigl\{T\in K(H):\|T\|_p=\Bigl(\sum_{n\ge1}s_n(T)^p\Bigr)^{1/p}<\infty\Bigr\}, \qquad S_\infty(H)=K(H). $$

Proposition (the classes are two-sided self-adjoint ideals). Each $S_p$ is a two-sided ideal of $B(H)$, self-adjoint under the involution, and

$$ S_1\subseteq S_p\subseteq S_q\subseteq K(H)\qquad(1\le p\le q<\infty), $$

with $S_1$ the trace class and $S_2$ the Hilbert–Schmidt class.

Proof. The inclusion of the classes is the comparison of $\ell^p$ norms on the singular values; the ideal property follows because the singular values of $ATB$ are dominated by $\|A\|\,\|B\|$ times those of $T$ (a standard majorisation), and $(S_p)^*=S_p$ because $|T^*|$ and $|T|$ are unitarily equivalent and have the same singular values.

$$ \|ST\|_r\le\|S\|_p\|T\|_q . $$

In particular $S_p S_q\subseteq S_r$, $B(H)S_p B(H)\subseteq S_p$, and the Hilbert–Schmidt norm is induced by the inner product

$$ \langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*), $$

under which $S_2(H)$ is a Hilbert space.

Proof. The triangle inequality for $\|\cdot\|_p$ is the majorisation form of the triangle inequality in $\ell^p$; the limiting statement is immediate from $s_n\to0$; Hölder's inequality for singular values gives the product estimate; the Hilbert–Schmidt inner product is linear in the first argument and conjugate-linear in the second by the properties of the trace.

Theorem (trace and duality). For $T\in S_1(H)$ the series $\operatorname{tr}T=\sum_n\langle Te_n,e_n\rangle$ converges absolutely and is independent of the orthonormal basis; $\operatorname{tr}(ST)=\operatorname{tr}(TS)$ for $S\in B(H)$, $T\in S_1(H)$; and the pairing $(S,T)\mapsto\operatorname{tr}(ST)$ identifies

$$ S_1(H)^*=B(H),\qquad K(H)^*=S_1(H),\qquad B(H)_*=S_1(H), $$

so $S_1(H)$ is the predual of $B(H)$ and the weak operator topology is the weak-* topology of this predual.

Proof. For $T\in S_1$ the series is absolutely summable by $|\langle Te_n,e_n\rangle|\le s_n$ and basis-independence is the standard trace computation; the cyclicity is the trace property; the duality $B(H)_*=S_1$ is the trace-class duality, and $K(H)^*=S_1$ is the same pairing restricted to the compact ideal. The statement that the WOT is the weak-* topology is then the identification of the rank-one trace-class functionals $\langle\cdot x,y\rangle$ with a total set of predual elements.

Remark. The chain $F(H)\subset S_1\subset S_2\subset K(H)\subset B(H)$ is strict in infinite dimension, and each class except $F(H)$ is closed. The Hilbert–Schmidt class is the one most often met: an integral operator on $L^2(X,\mu)$ with square-integrable kernel is Hilbert–Schmidt and hence compact, and every Hilbert–Schmidt operator is compact but the converse fails as the diagonal operator with $a_n=1/\log n$ shows.

Worked Cases

Integral Operators

For a measure space $(X,\mu)$ and a kernel $k\in L^2(X\times X)$ the operator

$$ (T_k f)(x)=\int_X k(x,y)f(y)\,d\mu(y) $$

is Hilbert–Schmidt on $L^2(X,\mu)$ with $\|T_k\|_{\mathrm{HS}}=\|k\|_2$ and $T_k^*=T_{\bar k}$ for the transposed kernel $\bar k(x,y)=\overline{k(y,x)}$; hence $T_k$ is compact, and it is self-adjoint exactly when $k(x,y)=\overline{k(y,x)}$ almost everywhere.

Diagonal Operators

On $\ell^2$ let $D_a e_n=a_n e_n$ with $a\in\ell^\infty$. Then $D_a$ is compact exactly when $a_n\to0$, in which case $s_n(D_a)$ is the non-increasing rearrangement of $|a_n|$, and $D_a\in S_p$ exactly when $(a_n)\in\ell^p$. So $S_1$ is the space of summable diagonals, $S_2$ of square-summable diagonals, and the modulus of a compact diagonal operator realises the absolute value of the symbol.

Finite Rank

In finite dimension every operator is compact, $K(H)=B(H)=S_1=S_2=F(H)$, the singular values are the eigenvalues of $|T|$, and $\|T\|_1=\operatorname{tr}|T|$ and $\|T\|_2$ are the trace and Frobenius norms of the matrix. The strict chain of the infinite-dimensional case collapses to a single class.

Summary

A compact operator on a Hilbert space is one whose image of the unit ball is relatively compact, equivalently a norm limit of finite-rank operators; the compact operators $K(H)$ form a closed two-sided self-adjoint ideal of $B(H)$, the unique proper closed two-sided ideal, and the quotient is the simple Calkin algebra $\mathcal{C}(H)$. The spectral theory of a compact operator is the Riesz–Schauder theory: the spectrum is countable, every nonzero spectral value is an eigenvalue of finite multiplicity, the only possible accumulation point is $0$, and a compact self-adjoint operator is diagonalised by an orthonormal eigenbasis of the Hilbert space. The singular values $s_n(T)$ organise the compact operators into the Schatten classes $S_p$, two-sided self-adjoint ideals Brun–Hölder-normed and nested between the finite-rank operators and the compact operators, of which the trace class $S_1$ and the Hilbert–Schmidt class $S_2$ are distinguished: $S_2$ is a Hilbert space under $\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$, and $S_1$ is the predual of $B(H)$ whose pairing with $B(H)$ realises the weak operator topology. Integral operators with square-integrable kernels are the standard examples, and the diagonal operators on $\ell^2$ show the whole chain $F(H)\subset S_1\subset S_2\subset K(H)\subset B(H)$ to be strict in infinite dimension.

Summary of Notation

Symbol	Meaning
$K(H)$	the compact operators, the unique proper closed ideal of $B(H)$
$F(H)$	the finite-rank operators, dense in $K(H)$
$\mathcal{C}(H)=B(H)/K(H)$	the Calkin algebra, simple
$\sigma(K)$	the spectrum; countable, nonzero points are eigenvalues, accumulation only at $0$
$s_n(T)$	singular values, eigenvalues of $\|T\|=(T^*T)^{1/2}$, non-increasing to $0$
$S_p(H)$, $\\|T\\|_p$	Schatten $p$-class and its norm; $S_\infty=K(H)$
$S_1$, $S_2$	trace class and Hilbert–Schmidt class
$\langle S,T\rangle_{\mathrm{HS}}=\operatorname{tr}(ST^*)$	Hilbert–Schmidt inner product
$B(H)_=S_1(H)$, $K(H)^=S_1(H)$	trace-class predual and duality
$\operatorname{tr}$	the trace, $\operatorname{tr}(ST)=\operatorname{tr}(TS)$