Positive definite kernel

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In operator theory, a positive definite kernel is a generalization of a positive matrix.

1 Definition
2 Examples
- 2.1 Reproducing kernel Hilbert space
- 2.2 Direct sum and tensor product
3 Characterization
- 3.1 Motivation
- 3.2 Kolmogorov decomposition
4 Some applications
- 4.1 Stinespring dilation theorem
5 See also
6 References

[edit] Definition

Let

$\{ H_n \}_{n \in {\mathbb Z}}$

be a sequence of (complex) Hilbert spaces and

$\mathcal{L}(H_i, H_j)$

be the bounded operators from H_i to H_j.

A map A on Z × Z where A(i, j) lies in

$\mathcal{L}(H_i, H_j)$

is called a positive definite kernel if for all m > 0 and h_i ε H_i, the following positivity condition holds:

$\sum_{-m \leq i,j \leq m} \langle A(i,j) h_j, h_i \rangle \geq 0.$

[edit] Examples

Positive definite kernels provide a framework that encompasses some basic Hilbert space constructions.

[edit] Reproducing kernel Hilbert space

Further information: Reproducing kernel Hilbert space

The definition and characterization of positive kernels extend verbatim to the case where the integers Z is replaced by an arbitrary set X. One can then give a fairly general procedure for constructing Hilbert spaces that is itself of some interest.

Consider the set F₀(X) of complex-valued functions f: X → C with finite support. With the natural operations, F₀(X) is called the free vector space generated by X. Let δ_x be the element in F₀(X) defined by δ_x(y) = δ_xy. The set {δ_x}_{x ∈ X} is a vector space basis of F₀(X).

Suppose now K: X × X → C is a positive definite kernel, then the Kolmogorov decomposition of K gives a Hilbert space

$({\mathcal H}, \langle \;, \; \rangle)$

where F₀(X) is "dense" (after possibly taking quotients of the degenerate subspace). Also, <[δ_x], [δ_y]> = K(x,y), which is a special case of the square root factorization claim above. This Hilbert space is called the reproducing kernel Hilbert space with kernel K on the set X.

Notice that in this context, we have (from the definition above)

$\{ H_n \}_{n \in {\mathbb Z}}$

being replaced by

$\{ {\mathbb C} \}_{x \in X}.$

Thus the Kolmogorov decomposition, which is unique up to isomorphism, starts with F₀(X).

One can readily show that every Hilbert space is isomorphic to a reproducing kernel Hilbert space on a set whose cardinality is the Hilbert space dimension of H. Let {e_x}_{x ∈ X} be an orthonormal basis of H The then kernel K defined by K(x, y) = <e_x, e_y> = δ_xy reproduces a Hilbert spaceH' . The bijection taking e_x to δ_x extends to a unitary operator from H to H' .

[edit] Direct sum and tensor product

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Let H(K, X) denote the Hilbert space corresponding to a positive kernel K on X × X. The structure of H(K, X) is encoded in K. One can thus describe, for example, the direct sum and the tensor product of two Hilbert spaces via their kernels.

Consider two Hilbert spaces H(K, X) and H(L, Y). The disjoint union of X and Y is the set

$X \sqcup Y = \{ (x, \xi)| x \in X\} \cup \{ (\xi, y)| y \in Y \} .$

Define a kernel

$K \oplus L$

on this disjoint union in a way that is similar to direct sum of positive matrices, and the resulting Hilbert space

$H( K \oplus L , X \sqcup Y)$

is then the direct sum, in the sense of Hilbert spaces, of H(K, X) and H(L, Y).

For the tensor product, a suitable kernel

$K \otimes L$

is defined on the Cartesion product X × Y in a way that extends the Kronecker product of positive matices:

$(K \otimes L) ((x,y), (x', y')) = K(x, x') L(y, y').$

This positive kernel gives the tensor product of H(K, X) and H(L, Y),

$H (K \otimes L, X \times Y)$

in which the family { [δ_(x,y)] } is a total set, i.e. its linear span is dense.

[edit] Characterization

[edit] Motivation

Consider a positive matrix A ∈ C^{n × n}, whose entries are complex numbers. Every such matrix A has a "square root factorization" in the following sense:

A = B*B where B: Cⁿ → H_A for some (finite dimensional) Hilbert space H_A.

Furthermore, if C and G is another pair, C an operator and G a Hilbert space, for which the above is true, then there exists a unitary operator U: G → H_A such that B = UC.

The can be shown readily as follows. The matrix A induces a denegerate inner product <·, ·>_A given by <x, y>_A = <x, Ay>. Taking the quotient with respect to the degenerate subspace gives a Hilbert space H_A, a typical element of which is an equivalence class we denote by [x].

Now let B: Cⁿ → H_A be the natural projection map, Bx = [x]. One can calculate directly that

$\langle x, B^*By\rangle = \langle Bx, By\rangle_A = \langle[x], [y]\rangle_A = \langle x, Ay\rangle$ .

So B*B = A. If C and G is another such pair, it is clear that the operator U: G → H_A that takes [x]_G in G to [x] in H_A has the properties claimed above.

If {e_i} is a given orthonormal basis of Cⁿ, then {B_i = Be_i} are the column vectors of B. The expression A = B*B can be rewritten as A_{i, j} = B_i*B_j. By construction, H_A is the linear span of {B_i}.

[edit] Kolmogorov decomposition

This preceding discussion shows that every positive matrix A with complex entries can expressed as a Gramian matrix. A similar description can be obtained for general positive definite kernels, with an analogous argument. This is called the Kolmogorov decomposition:

Let A be a positive definite kernel. Then there exists a Hilbert space H_A and a map B defined on Z where B(n) lies in

$\mathcal{L}(H_n, H_A) \quad \mbox{such that} \quad A(i,j) = B^*(i)B(j) \quad \mbox{and} \quad H_A = \bigvee_{n \in {\mathbb Z}} B(n) H_n \;.$

The condition that H_A = ∨B(n)H_n is referred to as the minimality condition. Similar to the scalar case, this requirement implies unitary freedom in the decomposition:

If there is a Hilbert space G and a map C on Z that gives a Kolmogorov decomposition of A, then there is a unitary operator

$U: G \rightarrow H_A \quad \mbox{such that} \quad UC(n) = B(n) \quad \mbox{for all} \quad n \in {\mathbb Z}.$

[edit] Some applications

[edit] Stinespring dilation theorem

Further information: Stinespring dilation theorem

Please help improve this section by expanding it.
Further information might be found on the talk page or at requests for expansion.

[edit] See also

Positive definite function on a group

[edit] References

D.E. Evans and J.T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. A, 24, 1977.

B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, 1970.

Categories: Operator theory | Hilbert space