Sentence examples for a continuous operator on from inspiring English sources

The symbol of a continuous operator on vector-valued white noise functionals is characterized by its analyticity and boundedness.

For instance, it is shown that if an infinite positive matrix A = [aij] defines a continuous operator on an lp-space (1 ≤ p < ∞) and A is quasinilpotent at a positive vector, then for any bounded double sequence of complex numbers {wij: i,j = 1, 2,... } the operator defined by the weighted infinite matrix [wijaij] has a non-trivial complemented invariant closed subspace.

Obviously, B is a continuous operator on B ̄ r ( 0 ).

This implies that T is a continuous operator on Ω.

Step 3. N is a continuous operator on (B_{l}).

Next, we show that H is a continuous operator on (overline {B 0,r_{0})}).

A continuous operator semigroup on is said to be uniformly asymptotically regular on if for all and any bounded subset of, we have (2.3).

Let H be a real Hilbert space, Γ : = { S ( t ) : 0 ≤ t < ∞ } be a continuous operator semigroup on H. Then Γ is said to be uniformly asymptotically regular (in short, u.a.r). on H if for all h ≥ 0 and any bounded subset C of H, lim t → ∞ sup x ∈ C ∥ S ( h ) ( S ( t ) x ) − S ( t ) x ∥ = 0.

Let C be a nonempty closed and convex subset of a Banach space X, S = T t : t > 0 be a continuous operator semigroup on C. Then  S is said to be uniformly asymptotically regular (in short, u.a.r). on C if for all h ≥ 0 and any bounded subset B of C such that lim t → ∞ sup x ∈ B T h T t x - T t x = 0.

Now it is shown, as in the proof of Theorem 3.1, that A is a Lipschitzian with the Lipschitz constant L and B is a completely continuous operator on [ a, b ].

(3.10) Since A is defined in (2.1) as a continuous operator that depends on y, for (f(Ay,y)) there exists a maximum for (y in[0,L]).