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In the context of kernel optimization, we prove a result that yields new factorizations and realizations. Our initial context is that of general positive operator-valued kernels. We further present implications for Hilbert space-valued…
Computing a consensus object from a set of given objects is a core problem in machine learning and pattern recognition. One popular approach is to formulate it as an optimization problem using the generalized median. Previous methods like…
The purpose of this article is to study operators whose kernel share some key features of Bergman kernels from complex analysis, and are approximate projectors. It turns out that they must be associated with a rich set of geometric data, on…
The generalization of the area theorem is derived for the case of a pulse circulating inside a ring laser cavity. In contrast to the standard area theorem, which is valid for a single pass of a traveling pulse through a resonant medium, the…
We review the basic properties of effective actions of families of theories (i.e., the actions depending on additional non-perturbative moduli along with perturbative couplings), and their description in terms of operators (called…
In this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions: a data is represented by a set of functions, and the label is also a function. We focus on the use of…
This paper considers paired operators in the context of the Lebesgue Hilbert space $L^2$ on the unit circle and its subspace, the Hardy space $H^2$. The kernels of such operators, together with their analytic projections, which are…
We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The…
Recent decades have provided a host of examples and applications motivating the study of nonlocal differential operators. We discuss a class of such operators acting on bounded domains, focusing on those with integrable kernels having…
We introduce the notions of kernel map and kernel set of a bounded linear operator on a Hilbert space relative to a subspace lattice. The characterization of the kernel maps and kernel sets of finite rank operators leads to showing that…
The purpose of this paper is to give an overview of the operator structure of frames, where the operator belongs to certain classes of linear operators and the element belongs to $H$. We discuss the size of the set of such elements. Also,…
A convenient technique for calculating completed topological tensor products of functional Frechet or DF spaces is developed. The general construction is applied to proving kernel theorems for a wide class of spaces of smooth and entire…
A general concept of a Hausdorff-type operator that absorbs all types of operators bearing the name `` Hausdorff operator'' and many others is considered. The characteristic features of this concept are the consideration of kernels…
We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation…
Here Lq-Lp boundedness of integral operator with operator-valued kernels is studied and the main result is applied to convolution operators. Using these results Besov space regularity for Fourier multiplier operator is established.
There are presented certain results on extending continuous linear operators defined on spaces of E-valued continuous functions (defined on a compact Hausdorff space X) to linear operators defined on spaces of E-valued measurable functions…
Motivated by practical applications, I present a novel and comprehensive framework for operator-valued positive definite kernels. This framework is applied to both operator theory and stochastic processes. The first application focuses on…
It was shown recently that the space isomorphic with an Gelfand Shilov space is well adapted for the use in quantum field theory with a fundamental length. It is our believe that all Gelfand Shilov spaces, especially those with…
We consider positive semidefinite kernels valued in the $*$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $*$-semigroups. For…
We study integral operators on the space of square-integrable functions from a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We establish…