We show that a formal power series in $2N$ non-commuting indeterminates is a positive non-commutative kernel if and only if the kernel on $N$-tuples of matrices of any size obtained from this series by matrix substitution is positive. We present two versions of this result related to different classes of matrix substitutions. In the general case we consider substitutions of jointly nilpotent $N$-tuples of matrices, and thus the question of convergence does not arise. In the ``convergent'' case we consider substitutions of $N$-tuples of matrices from a neighborhood of zero where the series converges. Moreover, in the first case the result can be improved: the positivity of a non-commutative kernel is guaranteed by the positivity of its values on the diagonal, i.e., on pairs of coinciding jointly nilpotent $N$-tuples of matrices. In particular this yields an analogue of a recent result of Helton on non-commutative sums-of-squares representations for the class of hereditary non-commutative polynomials. We show by an example that the improved formulation does not apply in the ``convergent'' case.
A connected homogeneous space X=G/K is called commutative if G is a connected Lie group, $K$ is a compact subgroup and the B*-algebra L^1(X)^K of K-invariant integrable function on X is commutative. In this article we introduce the space L^2_A (X) of A-bandlimited function on X by using the spectral decomposition of L^2 (X). We show that those spaces are reproducing kernel Hilbert spaces and determine the reproducing kernel. We then prove sampling results for those spaces using the smoothness of the elements in L^2_A (X). At the end we discuss the example of R^d, the spheres S^d, compact symme…
A classical theorem of Fatou asserts that the Radon-Nikodym derivative of any finite positive Borel measure, $μ$, with respect to Lebesgue measure on the complex unit circle, is recovered as the non-tangential limits of its Poisson transform in the complex unit disk. This positive harmonic Poisson transform is the real part of an analytic function whose Taylor coefficients are in fixed proportion to the conjugate moments of $μ$. Replacing Taylor series in one variable by power series in several non-commuting variables, we show that Fatou's Theorem and related results have natural extensions to…
First, Ornstein and Sucheston proved that for a given positive contraction $T:L_1\to L_1$ there exists $m\in N$ such that $\big\|T^{m+1}-T^m\|<2$ then $$ \lim_{n\to\infty}\|T^{n+1}-T^n\|=0. $$ Such a result was labeled as "zero-two" law. In the present paper, we prove a generalized uniform "zero-two" law for multi-parametric family of positive contractions of the non-commutative $L_1$-spaces. Moreover, we also establish a vector-valued analogous of the uniform "zero-two" law for positive contractions of $L_1(M,Φ)$-- the non-commutative $L_1$-spaces associated with center valued trace.
It is shown that a positive (bounded linear) operator on a Hilbert space with trivial kernel is unitarily equivalent to a Hankel operator that satisfies double positivity condition if and only if it is non-invertible and has simple spectrum (that is, if this operator admits a cyclic vector). More generally, for an arbitrary positive (bounded linear) operator A on a Hilbert space H with trivial kernel the collection V(A) of all linear isometries V from H into H such that AV is positive as well is investigated. In particular, operators A such that V(A) contains a pure isometry with a given defic…
By introducing a suitable new definition for the inner product on the octonionic Bergman space, we determine the explicit form of the octonionic Bergman kernel, in the framework of octonionic analysis which is non-commutative and non-associative.
We study properties of positive operators on Gelfand-Shilov spaces, and distributions which are positive with respect to non-commutative convolutions. We prove that boundedness of kernels $K \in \maclD_s^{\prime}$ to positive operators, are completely determined by the behaviour of $K$ alone the diagonal. We also prove that positive elements $a$ in $\mascS^{\prime}$ with respect to twisted convolutions, having Gevrey class property of order $s\geq 1/2$ at the origin, then $a$ belongs to the Gelfand-Shilov space $\maclS_s$.
The signature(s) of the curvature of the zero set V of a free (non-commutative) polynomial is defined as the number of positive and negative eigenvalues of the non-commutative second fundamental form on V determined by p. With some natural hypotheses, the degree of p is bounded in terms of the signature. In particular, if one of the signatures is zero, then the degree of p is at most two.
We study classes of reproducing kernels $K$ on general domains; these are kernels which arise commonly in machine learning models; models based on certain families of reproducing kernel Hilbert spaces. They are the positive definite kernels $K$ with the property that there are countable discrete sample-subsets $S$; i.e., proper subsets $S$ having the property that every function in $\mathscr{H}\left(K\right)$ admits an $S$-sample representation. We give a characterizations of kernels which admit such non-trivial countable discrete sample-sets. A number of applications and concrete kernels are …
In this paper we study positive fixed points of Hammerstein integral operators with degenerate kernel in the cone of C[0, 1]. Problem on a number of positive fixed points of the Hammerstein integral operator leads to the study positive roots of polynomials with real coefficients. Consider a model on a Cayley tree with nearest-neighbor interactions and with the set [0, 1] of spin values. The uniqueness translational-invariant Gibbs measures for the given model is proved.
We present a necessary and sufficient condition for the strict positive definiteness of a real, continuous, isotropic and positive definite kernel on a two-point compact homogeneous space. The characterization adds to others previously obtained by D. Chen at all (2003) in the case in which the space is a sphere of dimension at least 2 and Menegatto at all (2006) in the case in which the space is the unit circle. As an application, we use the characterization to improve upon a recent result on the differentiability of positive definite kernels on the spaces.
We extend results on analytic complex measures on the complex unit circle to a non-commutative multivariate setting. Identifying continuous linear functionals on a certain self-adjoint subspace of the Cuntz--Toeplitz $C^*-$algebra, the free disk operator system, with non-commutative (NC) analogues of complex measures, we refine a previously developed Lebesgue decomposition for positive NC measures to establish an NC version of the Frigyes and Marcel Riesz Theorem for `analytic' measures, i.e. complex measures with vanishing positive moments. The proof relies on novel results on the order prope…
We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb's results on maximizers of multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends several inverse inequalities.
We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present ageneral positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysisand metric geometry and provide a number of examples.
The Carathéodory problem in the $N$-variable non-commutative Herglotz--Agler class and the Carathéodory--Fejér problem in the $N$-variable non-commutative Schur--Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory--Fejér) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set $Λ$) takes operator values with positive semidefinite real part (resp., contractive operator values) on $N$-tuples of $Λ$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N…
We prove some inequalities for the spectral radius of positive operators on Banach function spaces. In particular, we show the following extension of Levinger's theorem. Let $K$ be a positive compact kernel operator on $L^2(X,μ)$ with the spectral radius $r(K)$. Then the function $ϕ$ defined by $ϕ(t) = r(t K + (1-t) K^*)$ is non-decreasing on $[0, {1/2}]$. We also prove that $\| A + B^* \| \ge 2 \cdot \sqrt{r(A B)}$ for any positive operators $A$ and $B$ on $L^2(X,μ)$.
We introduce a method to construct general multivariate positive definite kernels on a nonempty set $X$ that employs a prescribed bounded completely monotone function and special multivariate functions on $X$.\ The method is consistent with a generalized version of Aitken's integral formula for Gaussians.\ In the case where $X$ is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation.\ In addition, it can be interpreted as an abstract multivariate generalization of the well-established Gneiting's model for constructing …
A Positive Operator Valued Measure (POVM) is a map $F:\mathcal{B}(X)\to\mathcal{L}_s^+(\mathcal{H})$ from the Borel $σ$-algebra of a topological space $X$ to the space of positive self-adjoint operators on a Hilbert space $\mathcal{H}$. We assume $X$ to be Hausdorff, locally compact and second countable and prove that a POVM $F$ is commutative if and only if it is the smearing of a spectral measure $E$ by means of a Feller Markov kernel. Moreover, we prove that the smearing can be realized by means of a strong Feller Markov kernel if and only if $F$ is uniformly continuous. Finally, we prove t…
Let $(X,μ)$ be a strictly-positive Borel measure space. We show that the modes of convergence in a reproducing kernel Hilbert (RKHS) space, pointwise, weak and strong are all equivalents. From this we describe some important consequences such as an association with positive operators and positive definite kernels and a compact embedding condition for a RKHS in $L^{2}(X,μ)$.
We prove new inequalities and equalities for the generalized and the joint spectral radius (and their essential versions) of Hadamard (Schur) geometric means of bounded sets of positive kernel operators on Banach function spaces. In the case of nonnegative matrices that define operators on Banach sequences we obtain additional results. Our results extend results of several authors that appeared relatively recently.
The Riesz-Markov theorem identifies any positive, finite, and regular Borel measure on the complex unit circle with a positive linear functional on the continuous functions. By the Weierstrass approximation theorem, the continuous functions are obtained as the norm closure of the Disk Algebra and its conjugates. Here, the Disk Algebra can be viewed as the unital norm-closed operator algebra of the shift operator on the Hardy Space, $H^2$ of the disk. Replacing square-summable Taylor series indexed by the non-negative integers, i.e. $H^2$ of the disk, with square-summable power series indexed b…
Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the kernel can be recovered as a generalized convolution root of an equally positive definite and zonal kernel.
In this paper, we consider a (nonlinear) transformation $Φ$ of invertible positive elements in $C^*$-algebras which preserves the norm of any of the three fundamental means of positive elements; namely, $\|Φ(A)\mm Φ(B)\| = \|A\mm B\|$, where $\mm$ stands for the arithmetic mean $A\nabla B=(A+B)/2$, the geometric mean $A\#B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$, or the harmonic mean $A!B=2(A^{-1} + B^{-1})^{-1}$. Assuming that $Φ$ is surjective and preserves either the norm of the arithmetic mean or the norm of the geometric mean, we show that $Φ$ extends to a Jordan $*$-isomorphism between …
We investigate the notion of conditionally positive definite in the context of Hilbert $C^*$-modules and present a characterization of the conditionally positive definiteness in terms of the usual positive definiteness. We give a Kolmogorov type representation of conditionally positive definite kernels in Hilbert $C^*$-modules. As a consequence, we show that a $C^*$-metric space $(S, d)$ is $C^*$-isometric to a subset of a Hilbert $C^*$-module if and only if $K(s,t)=-d(s,t)^2$ is a conditionally positive definite kernel. We also present a characterization of the order $K'\leq K$ between condit…
We discuss the structure of positive definite kernels in terms of operator models. In particular, we introduce two models, one of Hessenberg type and another one that we call near triangular. These models produce parametrizations of the kernels and we describe the combinatorial nature of these parametrizations in terms of lattice paths of Dyck and Lukasiewicz type.
We present a characterization for a positive definite operator valued kernel to be universal or $C_{0}$-universal, and apply these characterizations to a family of operator valued kernels that are shown to be well behaved. Later, we obtain a characterization for an operator valued differentiable kernel to be $C^{q}$-universal and $C_{0}^{q}$-universal. In order to obtain such characterization and examples we generalize some well known results concerning the structure of differentiable kernels to the operator valued context. On the examples is given an emphasis on the radial kernels on Euclidea…
In this paper we consider non-commutative analogue for the arithmeticgeometric mean inequality $$a^{r}b^{1-r}+(r-1)b\geq ra$$ for two positive numbers $a,b$ and $r> 1$. We show that under some assumptions the non-commutative analogue for $a^{r}b^{1-r}$ which satisfies this inequality is unique and equal to $r$-mean. The case $0<r<1$ is also considered. In particular, we give a new characterization of the geometric mean.
The generalized state space of a commutative C*-algebra, denoted S_H(C(X)), is the set of positive unital maps from C(X) to the algebra B(H) of bounded linear operators on a Hilbert space H. C*-convexity is one of several non-commutative analogs of convexity which have been discussed in this context. In this paper we show that a C*-extreme point of S_H(C(X)) satisfies a certain spectral condition on the operators in the range of the associated positive operator-valued measure. This result enables us to show that C*-extreme maps from C(X) into K^+, the algebra generated by the compact and scala…
Hilbert-Schmidt independence criterion and distance covariance are methods to describe independence of random variables using either the Kronecker product of positive definite kernels or the Kronecker product of conditionally negative definite kernels. In this paper we generalize both methods by providing an independence criteria using a new concept, of positive definite independent kernels. We provide a characterization of the radial kernels that are positive definite independent on all Euclidean spaces and we present several examples.
We consider a kernel based harmonic analysis of "boundary," and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivated by) results and notions from classical harmonic analysis on the disk. Our positive definite kernels include those defined on infinite discrete sets, for example sets of vertices in electrical networks, or discrete sets which arise from sampling operations performed on positive definite kernels in a continuous setting. Below we give a summary of main conclusions in the paper: Starting with a given p…
In this paper, we give a new approach to the theory of strictly positive kernels. Our method is based on the structure of Fock spaces. As its applications, various examples of strictly positive kernels are given. Moreover, we give a new proof of the universal approximation theorem for the Gauss kernel.
The full Fock space over $\mathbb C ^d$ can be identified with the free Hardy space, $H^2 (\mathbb B ^d _\mathbb N)$ - the unique non-commutative reproducing kernel Hilbert space corresponding to a non-commutative Szegö kernel on the non-commutative, multi-variable open unit ball $\mathbb B ^d _\mathbb N := \bigsqcup _{n=1} ^\infty \left( \mathbb C^{n\times n} \otimes \mathbb C ^d \right) _1$. Elements of this space are free or non-commutative functions on $\mathbb B ^d _\mathbb N$. Under this identification, the full Fock space is the canonical non-commutative and several-variable analogue of…
We study representations of positive definite kernels $K$ in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for the most general given positive definite kernel, but are then subsequently specialized to the above mentioned applications. Given a positive definite kernel $K$ on $S\times S$ where $S$ is a fixed set, we first study families of factorizations of $K$. By a factorization (or representation) we mean a probability space $\left(B,μ\right)$ and an associated stoc…
We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.…
We describe several classes of holomorphic functions of positive real part on the unit ball; each is characterized by an operator-valued Herglotz formula. Motivated by results of J.E. McCarthy and M. Putinar, we define a family of weighted Cauchy-Fantappiè pairings on the ball and establish duality relations between certain pairs of classes, and in particular we identify the dual of the positive Schur class. We also establish the existence of self-dual classes with respect to this pairing, and identify some extreme points of the positive Schur class.
We give a new proof of a classical result which provides a one-to-one correspondence between positive definite radial kernels on a homogeneous tree and finite Borel measures on the interval $[-1,1]$. Our methods allow us to find a new characterisation in terms of positive trace-class operators on $\ell_2$. Furthermore, we extend both characterisations to finite products of homogeneous trees. The proof relies on a formula for the norm of radial Schur multipliers, in the spirit of Haagerup--Steenstrup--Szwarc, and a variation of the Hamburger moment problem.
Using special technique of expanding ratio of densities in an infinite series of polynomials orthogonal with respect to one of the densities, we obtain simple, closed forms of certain kernels built of the so called Al-Salam-Chihara (ASC) polynomials. We consider also kernels built of some other families of polynomials such as the so called big continuous q-Hermite polynomials that are related to the ASC polynomials. The constructed kernels are symmetric and asymmetric. Being the ratios of the densities they are automatically positive. We expand also reciprocals of some of the kernels, getting …
This work is motivated by the frequent occurrence of boundary value problems with various boundary conditions in the modeling of some problems in engineering and physical science. Here we propose a new technique to force the positive definite kernels such as some radial basis functions to satisfy the boundary conditions exactly. It can improve the applications of existing methods based on positive definite kernels and radial basis functions especially the pseudospectral radial basis function method for handling the differential equations with more complicated boundary conditions. The proposed …
