Any bounded operator that has finite
rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of
finite-rank operators in an infinite-dimensional setting. When is a
Hilbert space, it is true that any compact operator is a limit of finite-rank operators,[1] so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the
norm topology. Whether this was true in general for Banach spaces (the
approximation property) was an unsolved question for many years; in 1973
Per Enflo gave a counter-example, building on work by
Grothendieck and
Banach.[2]
The origin of the theory of compact operators is in the theory of
integral equations, where integral operators supply concrete examples of such operators. A typical
Fredholm integral equation gives rise to a compact operator K on
function spaces; the compactness property is shown by
equicontinuity. The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea of
Fredholm operator is derived from this connection.
Equivalent formulations
A linear map between two
topological vector spaces is said to be compact if there exists a neighborhood of the origin in such that is a relatively compact subset of .[3]
Let be normed spaces and a linear operator. Then the following statements are equivalent, and some of them are used as the principal definition by different authors[4]
there exists a
neighbourhood of the origin in and a compact subset such that ;
for any bounded sequence in , the sequence contains a converging subsequence.
If in addition is Banach, these statements are also equivalent to:
the image of any bounded subset of under is
totally bounded in .
If a linear operator is compact, then it is continuous.
Important properties
In the following, are Banach spaces, is the space of bounded operators under the
operator norm, and denotes the space of compact operators . denotes the
identity operator on , , and .
is a closed subspace of (in the norm topology). Equivalently,[5]
given a sequence of compact operators mapping (where are Banach) and given that converges to with respect to the
operator norm, is then compact.
Conversely, if are Hilbert spaces, then every compact operator from is the limit of finite rank operators. Notably, this "
approximation property" is false for general Banach spaces X and Y.[4]
if the range of is closed in Y, then the range of is finite-dimensional.[5][7]
If is a Banach space and there exists an
invertible bounded compact operator then is necessarily finite-dimensional.[7]
Now suppose that is a Banach space and is a compact linear operator, and is the
adjoint or
transpose of T.
For any , then is a
Fredholm operator of index 0. In particular, is closed. This is essential in developing the spectral properties of compact operators. One can notice the similarity between this property and the fact that, if and are subspaces of where is closed and is finite-dimensional, then is also closed.
If is any bounded linear operator then both and are compact operators.[5]
If then the range of is closed and the kernel of is finite-dimensional.[5]
For every the set is finite, and for every non-zero the range of is a
proper subset of X.[5]
Origins in integral equation theory
A crucial property of compact operators is the
Fredholm alternative, which asserts that the existence of solution of linear equations of the form
(where K is a compact operator, f is a given function, and u is the unknown function to be solved for) behaves much like as in finite dimensions. The
spectral theory of compact operators then follows, and it is due to
Frigyes Riesz (1918). It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either a finite subset of C which includes 0, or the spectrum is a
countably infinite subset of C which has 0 as its only
limit point. Moreover, in either case the non-zero elements of the spectrum are
eigenvalues of K with finite multiplicities (so that K − λI has a finite-dimensional
kernel for all complex λ ≠ 0).
An important example of a compact operator is
compact embedding of
Sobolev spaces, which, along with the
Gårding inequality and the
Lax–Milgram theorem, can be used to convert an
elliptic boundary value problem into a Fredholm integral equation.[8] Existence of the solution and spectral properties then follow from the theory of compact operators; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues. One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarily high vibration frequencies always exist.
The compact operators from a Banach space to itself form a two-sided
ideal in the
algebra of all bounded operators on the space. Indeed, the compact operators on an infinite-dimensional separable Hilbert space form a maximal ideal, so the
quotient algebra, known as the
Calkin algebra, is
simple. More generally, the compact operators form an
operator ideal.
For Hilbert spaces, another equivalent definition of compact operators is given as follows.
An operator on an infinite-dimensional
Hilbert space
is said to be compact if it can be written in the form
where and are orthonormal sets (not necessarily complete), and is a sequence of positive numbers with limit zero, called the
singular values of the operator. The singular values can
accumulate only at zero. If the sequence becomes stationary at zero, that is for some and every , then the operator has finite rank, i.e, a finite-dimensional range and can be written as
The bracket is the scalar product on the Hilbert space; the sum on the right hand side converges in the operator norm.
An important subclass of compact operators is the
trace-class or
nuclear operators, i.e., such that . While all trace-class operators are compact operators, the converse is not necessarily true. For example tends to zero for while .
Completely continuous operators
Let X and Y be Banach spaces. A bounded linear operator T : X → Y is called completely continuous if, for every
weakly convergentsequence from X, the sequence is norm-convergent in Y (
Conway 1985, §VI.3). Compact operators on a Banach space are always completely continuous. If X is a
reflexive Banach space, then every completely continuous operator T : X → Y is compact.
Somewhat confusingly, compact operators are sometimes referred to as "completely continuous" in older literature, even though they are not necessarily completely continuous by the definition of that phrase in modern terminology.
Examples
Every finite rank operator is compact.
For and a sequence (tn) converging to zero, the multiplication operator (Tx)n = tn xn is compact.
For some fixed g ∈ C([0, 1]; R), define the linear operator T from C([0, 1]; R) to C([0, 1]; R) by
That the operator T is indeed compact follows from the
Ascoli theorem.
More generally, if Ω is any domain in Rn and the integral kernel k : Ω × Ω → R is a
Hilbert–Schmidt kernel, then the operator T on L2(Ω; R) defined by
is a compact operator.
By
Riesz's lemma, the identity operator is a compact operator if and only if the space is finite-dimensional.[9]
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
ISBN978-1584888666.
OCLC144216834.
Renardy, M.; Rogers, R. C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics. Vol. 13 (2nd ed.). New York:
Springer-Verlag. p. 356.
ISBN978-0-387-00444-0. (Section 7.5)