Sentence examples for concerning the operator from inspiring English sources

This operator reveals a rich structure through its representation as a martingale, and we obtain new results concerning the operator norm ofSacting on the class of differential forms havingLpcoefficients.

Concerning the operator (1.5), we can note the sufficient conditions of the fact that its spectral radius is less than one Define the set and If there exists such that ( then on every finite interval the spectral radius of the operator defined by the formula (1.5) for is zero.

Concerning the operator (5), we can note that the inequality esssup t ≥ 0 | q i ( t ) | < 1, t ∈ [ 0, ∞ ), i = 1, …, n, is a sufficient condition that the spectral radii ρ ( S i ) of the operators S i is less than 1. Below we assume that this inequality is fulfilled.

Now we can see the most important intertwining relations concerning the operators (x^{2}), (Delta_{h}), (E_{mu}).

Concerning the operators S i we assume that S i : L ∞ → L ∞ are linear continuous Volterra operators and the spectral radius ρ ( S i ) of the operator S i is less than 1 for i = 1, …, n.

The investigation of the boundary value problems, concerning the operators in the form of the sum of squares of vector fields fulfilling Hörmander condition, has turned into the subject of several works, see [2 4].

In what follows we recall facts concerning the superposition operator which are drawn from [25].

For the reader's convenience, in the next lemma we recall some results of [15] and [8, 9] concerning the multiplication operator u ∈ W ∘ i 1, 2 → g u ∈ L 2 , (2.1).

Concerning the Nemytskii operator, in the continuous case, it is not true that if f is u.s.p. and ϕ is s.p., then t ↦ f ( t, ϕ ( t ) ) is s.p.

ws-compact if it is continuous and maps relatively weakly compact sets of (mathcal{D}) into relatively strongly compact ones of Y; ww-compact if it is continuous and maps relatively weakly compact sets of (mathcal{D}) into relatively weakly compact ones of Y. Next, we collect a few auxiliary facts concerning the superposition operator required in the sequel.

In a more general setting, we establish basic properties concerning the composition operators acting on Fock spaces associated with noncommutative varieties VP0(H ⊆[B(H n]1 generated by sets P0 of noncommutative polynomials in n indeterminates such that p(0)="0, p∈P0.