The full gyrokinetic electrostatic linearized Landau collision operator is calculated including the equilibrium operator, which represents the effect of collisions between gyrokinetic Maxwellian particles. First, the equilibrium operator describes energy exchange between different plasma species, which is important in multiple ion-species plasmas. Second, the equilibrium operator describes drag...

Krein space operator-theoretic methods are used to prove that the operator (sgn x) d is similar to a selfadjoint operator in the Hilbert space L2 (R) . Let L be a symmetric ordinary differential expression. Spectral properties of the operators associated with the weighted eigenvalue problem Lu = )wu have been studied extensively. When w is positive, this problem leads to a selfadjoint problem i...

In this paper we study the asymptotic behavior of the ground state energy E(R) of the Schrödinger operator PR = −∆ + V1(x) + V2(x−R), x, R ∈ IR, where the potentials Vi are small perturbations of the Laplacian in IR, n ≥ 3. The methods presented here apply also in the investigation of the ground state energy E(g) of the operator Pg = P + V1(x) + V2(gx), x ∈ X, g ∈ G, where Pg is an elliptic ope...

We characterize classes of linear maps between operator spaces E, F which factorize through maps arising in a natural manner via the Pisier vector-valued non-commutative L spaces Sp[E ∗] based on the Schatten classes on the separable Hilbert space l. These classes of maps can be viewed as quasi-normed operator ideals in the category of operator spaces, that is in noncommutative (quantized) func...

Also compact operators of scalar type in any Banach can be so expressed, provided, the sum (1) is allowed to be countably infinite. It is perhaps less often noted that any operator of scalar type can be expressed in the form (1), with the index j running over all positive integers. But, in general, the projections P. cannot be chosen pair-wise J disjoint, that is, the product of any distinct pa...

We prove that spectral projections of Laplace-Beltrami operator on the m-complex unit sphere E∆ S2m−1 ([0, R)) are uniformly bounded as an operator from H(S) to L(S) for all p ∈ (1,∞). We also show that the Bochner-Riesz conjecture is true when restricted to cylindrically symmetric functions on R × R. Suppose that L is a positive definite, self-adjoint operator acting on L(X,μ), where X is a me...

where F (x) is the distribution function of X. The expectation operator has inherits its properties from those of summation and integral. In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)]. 2. E(aX +...

In this paper, we introduce and study a generalized Yosida approximation operator associated to H(·, ·)-co-accretive operator and discuss some of its properties. Using the concept of graph convergence and resolvent operator, we establish the convergence for generalized Yosida approximation operator. Also, we show an equivalence between graph convergence for H(·, ·)-co-accretive operator and gen...

The aim of this short note is to reconsider and to extend a former result of K. Mochizuki [Moc76], [MN96] on the existence of the scattering operator for wave equations with small dissipative terms. Contrary to the approach used by Mochizuki we construct the wave operator explicitly in terms of the parametrix construction obtained by a (simplified) diagonalization procedure, cf. [Yag97]. The me...

We use elementary methods to compute the L-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk-Zuk [4]. The latter result was used by Grigorchuk-Linnell-Schick-Zuk [3] to pro...