My name is Jestine (34 years old) and my hobbies are Origami and Microscopy.

Here is my web site; http://Www.hostgator1centcoupon.info/ (support.file1.com) In operator theory, a branch of mathematics, a positive definite kernel is a generalization of a positive-definite matrix.

Definition

Let

${\displaystyle \{H_{n}\}_{n\in {\mathbb {Z} }}}$

be a sequence of (complex) Hilbert spaces and

${\displaystyle {\mathcal {L}}(H_{i},H_{j})}$

be the bounded operators from H_i to H_j.

A map A on ${\displaystyle {\mathbb {Z} }\times {\mathbb {Z} }}$ where

${\displaystyle A(i,j)\in {\mathcal {L}}(H_{i},H_{j})}$

is called a positive definite kernel if for all m > 0 and ${\displaystyle h_{i}\in H_{i}}$, the following non-negativity condition holds:

${\displaystyle \sum _{-m\leq i\quad \, \atop j\leq m}\langle A(i,j)h_{i},h_{j}\rangle \geq 0.}$

Examples

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

Reproducing kernel Hilbert space

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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

${\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )}$

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)

${\displaystyle \{H_{n}\}_{n\in {\mathbb {Z} }}}$

being replaced by

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

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. Then the kernel K defined by K(x, y) = ⟨e_x, e_y⟩ = δ_xy reproduces a Hilbert space H. The bijection taking e_x to δ_x extends to a unitary operator from H to H' .

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

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

Define a kernel

${\displaystyle K\oplus L}$

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

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

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

${\displaystyle K\otimes L}$

is defined on the Cartesian product X × Y in a way that extends the Schur product of positive matrices:

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

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

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

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

Characterization

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 degenerate 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

${\displaystyle \langle x,B^{*}By\rangle =\langle Bx,By\rangle _{A}=\langle [x],[y]\rangle _{A}=\langle x,Ay\rangle }$.

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}.

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

${\displaystyle {\mathcal {L}}(H_{n},H_{A})\quad {\mbox{such that}}\quad A(i,j)=B^{*}(i)B(j)\quad {\mbox{and}}\quad H_{A}=\bigvee _{n\in {\mathbb {Z} }}B(n)H_{n}\;.}$

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

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

Some applications

Stinespring dilation theorem

Template:Further2

See also

• Positive definite function on a group

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.

Summary

DescriptionSingular perturbation convergence.jpg	English: A demonstration of convergence of singular perturbation solutions. Shows solutions to ${\displaystyle \epsilon y''+(1+\epsilon )y'+y=0,\,}$ with ${\displaystyle y(0)=0}$, ${\displaystyle y(1)=1}$, for varying ${\displaystyle \epsilon }$. I plotted it in Octave and then hand-edited the EPS file and converted to JPG. Roy W. Wright 02:46, 25 March 2007 (UTC)
Date	25 March 2007 (original upload date)
Source	Transferred from en.wikipedia to Commons.
Author	Roy W. Wright at English Wikipedia

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