Transporting positive definiteness

This is a rough guide to the topic which might be worthy to develop further on. Expanding positive definiteness beyond its presupposed scope is intriguing due to a number of possible applications. The paper though looking at the first glance a little bit sketchy may provide a basis for further research.

A little piece of history 1. The outstanding Béla Szőkefalvi-Nagy appendix [17] concerns Hilbert space operator valued functions defined and positive definite on * -semigroups. More precisely and with a somehow neutral notation, let S be a * -semigroup (an involution semigroup alternatively); a map ϕ : S → B(H) is said to be positive definite if It is said to satisfy the boundedness condition 1 if for every u ∈ S there is c(u) > 0 such that The above makes immediately ϕ(s)f, g H = Φ(s)V f, V g K , s ∈ S, f, g ∈ H.
which is sometimes referred to as a weak dilation; the other described above is just a dilation.
If ϕ(1) = I then V is an isometric embedding of H into K. Identifying H with its image V H in K allows us to think of V * in (3) as an orthogonal projection 2 ; this is in the flavour [17].
Notice properly understood minimality forces uniqueness to hold.
2. The content of [3,4] encouraged us to proclaim in [8] the following result 3 and only if (4) holds for s ∈ S.
3. Let us supply with a resume of the RKHS story (the long established reference is [1], the assuming is [16]). Given a set X, a function of two variables Denote by H a Hilbert space of complex valued functions on X. We will say that the couple (K, H) enjoys the reproducing property, if the sections which have to be called kernel functions, enjoy the following two properties Condition (5) ensures the reproducing property (6) to be executable.
Formulae (5) and (6) are trivially equivalent to The members of the couple (K, H) are tied one to the other: • H can be obtain from K via the Moore-Aronszajn completion procedure of D, this can be made possible because due to (6) D is dense in H; • (6) gives us immediately continuity of the functional determining the kernel functions K x by the Riesz representation theorem. Now (7) turns out into the definition of K. Notice by the way (5) implies positive definiteness of K and the above makes these two counterparts equivalent which rounds the whole story up. In other words, we have

Focal results
4. The introductory stuff in 1 sets our main point forth.
Suppose we are given k : X → C. Denote by A k a set of mappings 4 α : X → X such that there is a c α for which Notice k is a fixed parameter the whole construction depends on.
It is clear that from (8) we infer that K is a positive definite kernel on X. Denote by H K the corresponding reproducing kernel Hilbert space.
Due to (7) and (8) defines a linear functional on H K , which is continuous with the norm not exceeding c

5.
Applying the Riesz representation theorem to (9) we get existence of a function such that We can declare 6 now getting a kernel K (k) which is positive definite on A k . Denoting by H (k) the corresponding reproducing kernel Hilbert space, composed of complex functions on A k , we arrive at what we mean here by transporting the positive definite structure of the reproducing couple (K, H) to (K (k) , H (k) ).
Summing the above up we state the following Theorem 3. Given the above data satisfying in particular (8) we conclude in getting a positive definite kernel K (k) defined by (12) which in turn generates the inner product in H (k) , the corresponding reproducing kernel Hilbert space. For the functions κ α , α ∈ A k standing for kernel functions of (K (k) , H (k) ), after employing Remark 2 and formula (12) we reach and Theorem 3 becomes, what we call in (the first appearance is in [11]) the RKHS test. It settles when a complex function on X, in this case k belongs to H. This can be deduced directly using (10) and (11).
It may be interesting to notify 5 The notation • stands apparently for composition of functions. 6 The right hand side of (12) makes sense because of (10).
and suppose the kernel K is diagonal, that is This implies H = 2 (X, μ) with the orthonormal basis 8 K x = μ 1 2 x , x ∈ X, and the measure μ designated as μ({x}) = μ −1 x .
Remark 6. Supposing A k and X are disjoint 9 by f 1 ∪ f 2 we mean the function (call it union) defined on A k ∪ X which agree with f 1 and f 2 on A k and X respectively. For kernels, the procedure goes as follows. Let K 1 be a kernel on A k and K 2 that on X, the union K 1 ∪ K 2 of these kernel is defined as otherwise.
If K 1 and K 2 are positive definite, then so is the kernel K 1 ∪ K 2 and H K1∪K2 = H K1 ∪ H K2 . This is another way than (14) of glueing the spaces clolin H {κ α : α ∈ A (k) } and H. 6. Come back to the case when the function ϕ is operator valued like in (4). We appeal here to the environment provided by Pedrick's approach (see [6] and even better [15]; here we follow the exposition in [15].) Duality as it appears in Functional Analysis, cf. [ which is separating in a sense that B(f, g) = 0 for all g ∈ F =⇒ f = 0, 7 This is a RKHS due to the criterion involving evaluation functionals. 8 { x}x∈X is the "zero-one" orthonormal basis in 2 (X). 9 This does not happen in the case of Theorem 1 The spaces E and F when accompanied by B are referred to as being in duality.
As the spaces are complex and the Hilbert space is highlighted we prefer to impose B to be Hermitian bilinear 10 . The most recognised examples are • E is a linear space and F = E * , where E * is the algebraic dual of E, that is the space of all linear functionals on E; • E is a locally convex space and F = E , where E is the topological dual of E, that is the space of all continuous linear functionals on E the bilinear form in both these cases is just the standard pairing, that is B = · , − ; from now on we use the latter instead of B. Even more, we replace "duals" by "antiduals", the latter comes from the previous by taking the complex conjugates of the values of functionals in question which results in B supposed to be Hermitian bilinear. This is synchronised with the conclusion the classical F. Riesz representation theorem which in fact determines such kind of "duality" when establishing isomorphism between functionals (the topological dual of a Hilbert space) and the elements represented by them.
Suggestive enough E, F is a shorthand notation of the triplet (E, F, · , − ). Incidentally we abridge antiduality to duality hoping no confusion arises.
(Anti)duality is encoded in The first of these two itemised cases can always be directed in the second by introducing the so called σ(E, F) topology in E or reversing the role the spaces play a σ(F, E) topology in F; in general they are not the only possibilities.

7.
Pedrick's device is to turn a kernel of (general or abstract) values into a scalar kernel like those in subsections 4 and 5 (this is what he calls "tilde correspondence"). Then after performing specific tasks in this friendly scalar territory the way back (that is the reciprocal of tilde correspondence, which is always possible) establishes the wanted properties. Let us scrutinise this now.
The following items are the main objects in [15, cf. p.16] • two complex linear spaces E and F being in antiduality, the Hilbert spaces H E ⊂ E X and H F ⊂ F X which are composed of E-valued or F-valued resp. functions on X, • a family of linear operators K(x, y) : F → E, x, y ∈ X, which may be also recollected as 11 K : X × X → L(F, E) and which in turn let K be thought of as L(F, E)-valued kernel on X. 10 Instead of being linear in the second variable it is linear conjugate; call B to be Hermitian bilinear and the spaces E and F being in antiduality. 11 L(X , Y) denotes the totality of all linear operators from X into Y, which by the way is a complex linear space; as always shorten L(X , X ) to L(X ).
Consider G ⊂ E X and assume the couple (K, G) enjoys the reproducing property on X, that is the functions K x,f def = K(·, x)f are in G for any x ∈ X, f ∈ F (15) and This intensionally turns out K to be an L(F, E)-valued positive definite kernel. Positive definiteness means here Notice the formula (16) is just the reproducing property as adjusted to the current circumstances.
8. An important consequence of the reproducing formula (16), when combined with (15), is which is a counterpart of (7). Now we can offer the following definitions Remark 7. The deal is now with the complex valued kernel K. Because K is linear in the second variable and antilinear in the fourth, formula (17) ensures its positive definiteness getting now the form Therefore K has its reproducing kernel Hilbert space H composed of complex functions on X def = X × F. If there is any need to consider other duality than the standard topological one (E, E ) mentioned in subSection 6, it can be done without any further complication.
The main point is to check whether G is just the RKHS partner of K.
where · G stands for the norm in G. Consequently, ( K, G) is a reproducing couple.
This pave the way between (K, G) and ( K, G) endowing Pedrick's "tilde correspondence" with the reputation of a unitary map. Notice that the inverse of Pedrick's "tilde correspondence" can be determined by applying formula (18) due to the fact that the duality E, F is separating.

Remark 9.
Thinking of E = F = Hilbert space we jump into the situation considered in [17] and [9]. More general context is when a Banach space appears instead; this what some people may like.
we get an L(F, E)-valued kernel on N which due to (21) is positive definite in the sense of (17). Pedrick's tilde correspondence procedure makesã m+n = K(m, n) a scalar valued kernel (a sequence in fact) matured to be a Hamburger moment sequence having an integral representation in the usual sense. Reversing the tilde correspondence we come from this classical stuff (after some work) to the representation a n = R t n M (dt), where M is an L(F, E)-valued measure (its σ-additivity being in the strong operator topology in F) with the appropriate meaning of the integral.
10. The current developments can be perfectly illustrated by the choice of (20) to be scalar valued. This was done successfully in [12] and [13]; it is a need to mention that an attempt made some time ago is in [18].

Concluding remark
As mentioned in Abstract this is a starting point for the matters. As the reproducing kernel technique is deeply engaged in these investigations the lack of appropriate sources (except those not accessible for this or another reason like [14] or [16]) limit them for the time being. Nevertheless, we hope to be able to continue the issue in the desired direction.