In these notes we provide an introduction to compact linear operators on Banach and Hilbert spaces. These operators behave very much like familiar finite dimensional matrices, without necessarily having finite rank. For more thorough treatments, see [RS, Y]. Definition 1 Let X and Y be Banach spaces. A linear operator C : X → Y is said to be compact if for each bounded sequence {xi}i∈IN ⊂ X , t...

Let G be a step two nilpotent Lie group. In this paper, we give necessary and sufficient conditions on the operator valued symbols σ such that associated pseudo-differential operators Tσ are in class of Hilbert-Schmidt operators. As key to prove this, define (μ,ν)-Weyl transform derive trace formula for with L2(R2n). We show L2(G) same as symbol L2(R2n+r+k×R2n+r+k). Further, present characteriz...

Spectral analysis is performed on the Born equation, a strongly singular integral equation modeling interactions between electromagnetic waves and arbitrarily shaped dielectric scatterers. Compact Hilbert--Schmidt operator polynomials are constructed from Green of scattering scatterers with smooth boundaries. As consequence, it shown that has discrete spectrum, spectral series $ \sum_\lambda|\l...

in this paper, we investigate duality of modular g-riesz bases and g-riesz basesin hilbert c*-modules. first we give some characterization of g-riesz bases in hilbert c*-modules, by using properties of operator theory. next, we characterize the duals of a giveng-riesz basis in hilbert c*-module. in addition, we obtain sucient and necessary conditionfor a dual of a g-riesz basis to be again a g...

We study the Connes spectral distance of quantum states and analyse nonlocality a 4D generalized noncommutative phase space. By virtue Hilbert–Schmidt operatorial formulation, we obtain Dirac operator construct triple corresponding to Based on ball condition, some constraint relations about optimal elements, then calculate between two Fock states. Due noncommutativity, distances in spaces are s...

In this work, we introduce a theory of stochastic integration with respect to symmetric α-stable cylindrical Lévy processes. Since processes do not enjoy semi-martingale decomposition, our approach is based on decoupling inequality for the tangent sequence Radonified increments. This enables us characterise largest space predictable Hilbert-Schmidt operator-valued which are integrable an proces...

In this paper, we study the relationship between the acquisition of time-harmonic seismic data and the Dirichlet-to-Neumann map for the Helmholtz equation in dimension n ≥ 3. This relationship is established through the introduction of a single-layer potential operator. We analyze its properties with a view to so-called iterative full waveform inversion based on the Hilbert-Schmidt norm, that i...