Sentence examples for a dense range from inspiring English sources

They grouped themselves with the overachievers, immersed in a dense range of activities.

By Lemma 2.1, Δ a, b − λ I has not a dense range.

T has a dense range if and only if T ∗ is one to one.

Notice that the assumptions of Corollary 3.4 are met whenever T is a compact operator with a dense range.

An important example of a strongly compact algebra is the commutant of a compact operator with a dense range.

[8], p. 59. T has a dense range if and only if (T^{ast}) is 1-1.

For FTIR experiments focusing in on the physiologically relevant range of glucose concentrations (50-400 mg/dL) in aqueous and serum solutions, a denser range of nominal concentration values was used.

For eachϕ∈Hom K1(A), R) with dense range inRwe construct a closed derivationδinAwhich generates a one-parameter automorphism groupαofAsuch thatτ(δ u) u* =2πiϕ([u]) for any unitaryu∈D.

Let H be a complex, separable, infinite dimensional Hilbert space, and let (L(H)) denote the algebra of all linear bounded operators on H, (C(H)) the set of linear operators A with domain (D(A)) dense in H and range (R A)) contained in H and a graph (G A)) closed in (Htimes H). (K(H)) is the set of compact elements of (L(H)).

As another application of the construction of derivations, we show that ifAis aC*-algebra of the above type andα∈HInn(A) has the Rohlin property and comes fromϕ∈Hom K1(A), R) with dense range as in Kishimoto and Kumjian (preprint), then the crossed productA×αZis again of the same type; in particularA×αZis an AT algebra.

We show that if is a -quasihyponormal operator and is a -hyponormal operator, and if, where is a quasiaffinity (i.e., a one-one map having dense range), then is a normal and moreover is unitarily equivalent to.