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Positive Definite Kernel
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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 ''Hi'' to ''Hj''.

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, the following positivity condition holds:

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


EXAMPLES


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


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 ''F''0(''X'') of complex-valued functions ''f'': ''X'' → ''C'' with finite support. With the natural operations, ''F''0(''X'') is called the free vector space generated by ''X''. Let ''δx'' be the element in ''F''0(''X'') defined by ''δx''(''y'') = ''δxy'' . The set {''δx''}''x'' ∈ ''X'' is a vector space basis of ''F''0(''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 \;, \; angle)

where ''F''0(''X'') is "dense" (after possibly taking quotients of the degenerate subspace). Also, < [''δy'' > = ''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 ''F''0(''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 {''ex''}''x ∈ X'' be an orthonormal basis of ''H'' The then kernel ''K'' defined by ''K''(''x'', ''y'') = <''ex'', ''ey''> = ''δxy'' reproduces a Hilbert space''H' ''. The bijection taking ''ex'' to ''δx'' extends to an unitary operator from ''H'' to ''H' ''.


Direct sum and tensor product


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



The condition that ''HA'' = ∨''B''(''n'')''Hn'' 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 an unitary operator

:U: G ightarrow 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



SEE ALSO




REFERENCES