Sentence examples for bounded and the operator from inspiring English sources

A map is said to be -compact in if is bounded and the operator is compact.

A map N : Ω ¯ → Y is called L-compact in Ω ¯ if Q N is bounded and the operator K ( I − Q ) N : Ω ¯ → X is compact.

In view of, we have Since the set is bounded and the operator is compact, we know that the set is relatively compact, which implies that there exists a subsequence such that (3.18).

A map N : Ω ¯ → Y is said to be L-compact in Ω ¯ if Q N is bounded and the operator K ( I − Q ) N : Ω ¯ → X is compact.

Let Ω be an open bounded subset of X with (D(L capOmeganeqemptyset). A map (N:overline{Omega}rightarrow Y) is said to be L-compact in Ω̅ if (QN overline{Omega})) is bounded and the operator (K I-Q N:overline{Omega}rightarrow X) is compact.

Let Ω be an open bounded subset of X with (D(L capOmeganeqemptyset). A map (N:overline{Omega}rightarrow Y) is said to be L-compact in (overline{Omega}) if (Q_{1}N overline{Omega})) is bounded and the operator (L_{P_{1}}^{-1} I-Q_{1})N:overline{Omega}rightarrow X) is compact.

First, we consider the case when one of the operators is bounded and the other one belongs to the Hilbert Schmidt class.

Let (f:[m,M]tomathbb{R}) be a twice differentiable function with (f([m,M] subseteq U), (g:[m_{Phi (A)},M_{Phi(A)}]tomathbb{R}) be continuous with (g([m_{Phi (A)},M_{Phi(A)}])subseteq V), and (F Utimes Vto mathbb{R}) be bounded and operator monotone in the first variable.

Then: (a) If T is bounded

If T is bounded

Conversely, assume that the operator is bounded, and condition (3.31) holds.