We consider positive semidefinite kernels which have values given by bounded linear operators on certain bundles of Hilbert spaces and which are invariant under actions of $*$-semigroupoids. For these kernels, we prove that there exist generalised $*$-representations of the $*$-semigroupoids on the underlying reproducing kernel Hilbert spaces or, equivalently, on the underlying minimal linearisations, we characterise when the $*$-representations are performed by means of bounded operators and show that this always happens for inverse semigroupoids. Then, we consider Hermitian kernels which have values given by bounded linear operators on certain bundles of Hilbert spaces and which are invariant under actions of $*$-semigroupoids. Only those Hermitian kernels having certain boundedness properties can produce reproducing kernel Krein spaces but uniqueness is more complicated. However, for these kernels, generalised $*$-representations can be obtained. If $*$-representations with bounded linear operators are requested, then stronger boundedness conditions on the kernels are needed.
Positive semidefinite scalar kernels have been considered since the beginning of the twentieth century, in connection with partial differential equations by S. Zaremba [94], integral equations by J. Mercer [66], spaces of analytic functions by S. Szegö [91], S. Bergman [10], S. Bochner [12], general analysis questions by E.H. Moore [70], stochastic analysis by N. Kolmogorov [51], until two general formalisations were performed by N. Aronszajn [3] and L. Schwartz [87]. Then, this theory was extended to operator valued kernels, firstly by G. Pedrick [82] to locally convex spaces, then by R.M. Loynes [63], [64], [65], to vector valued Hilbert spaces, and so on. In the meantime, this theory was used in many other domains of mathematics, such as: the theory of group representations by M.A. Naimark [73], [74], R. Godement [43], S. Bochner [13], M.G. Krein [53], [54], M.B. Bekka and P. de la Harpe [9], function spaces by L. de Branges [15], canonical models in quantum mechanics by L. de Branges and J. Rovnyak [17], dilation theory by W.F. Stinespring [89], B. Sz.-Nagy [92], D.E. Evans and J.T. Lewis [34], T. Constantinescu [20], probability theory by K.R. Parthasarthy [77], [78], and K.R. Parthasarathy and K. Schmidt [79]. A partial image of the big picture on the theory and the many applications of the theory of positive semidefinite kernels can be seen in the monographs of A. Berlinet and C. Thomas-Agnan [11], V.R. Paulsen and M. Raghupathi [81], and S. Saitoh and Y. Sawano [84].
A very fruitful research direction in the theory of kernels was opened by the seminal series of five articles of M.G. Krein and H. Langer [55], [56], [57], [58], [59], in which Hermitian scalar kernels that are not positive semidefinite have been considered in connection with generalisations of the classical kernels of C. Caratheodory, G. Szegö, and R. Nevanlinna. These investigations triggered further research for the special case of Pontryagin spaces, where the existence and uniqueness hold very similar to the case of Hilbert spaces, as can be seen in the monograph of D. Alpay, A. Dijksma, J. Rovnyak, H. de Snoo [2]. For the more general case of Krein spaces, the situation is technically rather different.
Here, it should be pointed out that, in the seminal paper of L. Schwartz [87], reproducing kernel spaces, in his special interpretation of Hilbert spaces continuously contained in a given quasi-complete locally convex space, have been generalised to Krein spaces, which he called Hermitian spaces. A related fact concerning realisations of Hermitian analytic kernels as operator colligations, also called transfer functions, in Krein spaces was studied by A. Dijksma, H. Langer, and H.S.V. de Snoo in [31]. D. Alpay [1] showed that analiticity of the kernel is a sufficient condition for existence of the reproducing kernel Krein space, while, in general, as can be seen at L. Schwartz [87] and in the series of articles of T. Constantinescu with the author [21], [22], [23], [24], additional boundedness conditions are necessary. The result of Alpay was generalised to several complex variables in [25].
In the last twenty years, positive semidefinite kernels and their reproducing kernel Hilbert spaces were found very useful in machine learning, as can be seen, for example, in the monographs of I. Steinwart and C. Christman [90] and that of B. Schölkopf and A.J. Smola [85]. Then, operator valued kernels turned out to be extremely useful in learning theory [18], [19], [41], [45], [46], [47], [48], [49], [50], [60], [67], [69], and many others. Moreover, recently, Hermitian kernels and their reproducing kernel Krein spaces have been found useful in machine learning [75], [76]. It is understood for some good time that, kernels that are invariant under actions of certain semigroups or groups turned out to cover many other mathematical problems and were found to be useful in many applications, as pointed out in [9], [34], [79], [20], [24], [6], [7], [8].
In view of the recent interest in the theory of groupoids in functional analysis, which was triggered by the investigations of J. Cuntz and W. Krieger [27], operator valued positive semidefinite maps on * -semigroupoids have been considered by B. Udrea and the author in [42]. It is known that operator valued kernels that are invariant under left actions of * -semigroups contain many other problems and it is expected that this feature be preserved when * -semigroupoids are used instead of * -semigroups. The concept of partial dynamical system of R. Exel [35] might be related with actions of semigroupoids. In [41], the author and C. Tilki pointed out that certain applications of operator valued kernels to machine learning requires that the operator valued kernels be localised, that is, given a base set X of inputs, to each x ∈ X one should consider a Hilbert space H x that depends on x. Consequently, one has to consider Hilbert space bundles over the input set X which, in general, are not trivial in
This content is AI-processed based on open access ArXiv data.