Rn×Rn S( x+ y 2 , η) e2iπ〈x−y, η〉 u(y) dy dη : such a linear operator extends as a continuous operator from S ′(Rn) to S(R) while, in the case when S ∈ S ′(Rn ×R) , one can still define Op(S) as a linear operator from S(R) to S ′(Rn) ; also, Op sets up an isometry from L(R×R) onto the space of Hilbert-Schmidt operators on L(R) . The sharp composition S1#S2 of two symbols, say lying in S(R×R) , ...

We propose an independence criterion based on the eigenspectrum of covariance operators in reproducing kernel Hilbert spaces (RKHSs), consisting of an empirical estimate of the Hilbert-Schmidt norm of the cross-covariance operator (we term this a Hilbert-Schmidt Independence Criterion, or HSIC). This approach has several advantages, compared with previous kernel-based independence criteria. Fir...

The goal of this paper is to establish the applicability of the Lyapunov-Schmidt reduction and the Centre Manifold Theorem for a class of hyperbolic partial differential equation models with nonlocal interaction terms describing the aggregation dynamics of animals/cells in a one-dimensional domain with periodic boundary conditions. We show the Fredholm property for the linear operator obtained ...

Composition operators Cφ on the Hilbert Hardy space H 2 over the unit disk are considered. We investigate when convergence of sequences {φn} of symbols, (i.e., of analytic selfmaps of the unit disk) towards a given symbol φ, implies the convergence of the induced composition operators, Cφn → Cφ . If the composition operators Cφn are Hilbert–Schmidt operators, we prove that convergence in the Hi...

The spectral problem (A + V (z))ψ = zψ is considered with A, a self-adjoint operator. The perturbation V (z) is assumed to depend on the spectral parameter z as resolvent of another self-adjoint operator A ′ : V (z) = −B(A ′ − z) −1 B *. It is supposed that the operator B has a finite Hilbert-Schmidt norm and spectra of the operators A and A ′ are separated. Conditions are formulated when the p...

Abstract We study a Sturm–Liouville-type operator with operator-valued coefficients and its iso-spectral deformations, related to new two-component Camassa–Holm-type completely integrable dynamical systems. Based on specially devised gradient-holonomic scheme, generalizing the one before developed for studying spectral problem spatially multidimensional Hilbert–Schmidt Hilbert space, we constru...

In this work we introduce a theory of stochastic integration for operator-valued integrands with respect to some classes cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed via the radonification martingales by Hilbert-Schmidt operator theorem and unifies several other theories particular, our covers space valued L\'{e}vy process (which not required satisfy any...

We continue the spectral analysis of differential operators with complex coefficients, extending some results for Sturm-Liouville to higher order operators. give conditions essential spectrum be empty, and operator have compact resolvent. Conditions are given on coefficients resolvent Hilbert-Schmidt. These new even real i.e., selfadjoint case. Asymptotic is a central tool. See also https://ejd...

In this article, we provide an expansion (up to the fourth order of coupling constant) energy ground state Hamiltonian a quantum mechanical particle moving inside parabolic well in x-direction and constrained by presence two-dimensional impurity, modelled attractive isotropic Gaussian potential. By investigating associated Birman–Schwinger operator exploiting fact that such integral is Hilbert–...