Related papers: On Integro-Differential Inclusions with Operator-v…
We study linear integro-differential equations in Hilbert spaces with operator-valued kernels and give sufficient conditions for the well-posedness. We show that several types of integro-differential equations are covered by the class of…
A class of non-autonomous differential inclusions in a Hilbert space setting is considered. The well-posedness for this class is shown by establishing the mappings involved as maximal monotone relations. Moreover, the causality of the so…
We study an abstract class of autonomous differential inclusions in Hilbert spaces and show the well-posedness and causality, by establishing the operators involved as maximal monotone operators in time and space. Then the proof of the…
We consider an abstract class of differential inclusions, which covers differential-algebraic and non-autonomous problems as well as problems with delay. Under weak assumptions on the operators involved, we prove the well-posedness of those…
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…
This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace.…
We study the correct solvability of an abstract functional differential equations in Hilbert space, which includes integro-differential equations describing evolution of thermal phenomena, heat transfer in materials with memory or sound…
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…
To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are…
Much recent work has addressed the solution of a family of partial differential equations by computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert…
Motivated by applications, we introduce a general and new framework for operator valued positive definite kernels. We further give applications both to operator theory and to stochastic processes. The first one yields several dilation…
This paper presents a mathematical foundation for physical models in nonlinear optics through the lens of evolutionary equations. It focuses on two key concepts: well-posedness and exponential stability of Maxwell equations, with models…
In this article, we show that a technique for showing well-posedness results for evolutionary equations in the sense of [13] established in [16] applies to a broader class of non-autonomous integro-differential-algebraic equations. Using…
In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are…
In this paper, we present the general theory of embedding independence tests on Hilbert spaces that generalizes the concepts of distance covariance, distance multivariance and HSIC. This is done by defining new types of kernel on an $n$…
In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…
We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics,…
The difficulty for solving ill-posed linear operator equations in Hilbert space is reflected by the strength of ill-posedness of the governing operator, and the inherent solution smoothness. In this study we focus on the ill-posedness of…
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…
Functional analysis, especially the theory of Hilbert spaces and of operators on these, form an important area in mathematics. We formalized the Isabelle/HOL library Complex_Bounded_Operators containing a large amount of theorems about…