Compactness and weak-star continuity of derivations on weighted convolution algebras

TY - JOUR

T1 - Compactness and weak-star continuity of derivations on weighted convolution algebras

AU - Pedersen, Thomas Vils

PY - 2013/1/1

Y1 - 2013/1/1

N2 - Let ω be a continuous weight on R+ and let L 1(ω) be the corresponding convolution algebra. By results of Grønbæk and Bade & Dales the continuous derivations from L 1(ω) to its dual space L ∞(1/ω) are exactly the maps of the form (D φf)(t)=∫ 0 ∞f(s)s/t+sφ(t+s)ds (t∈R{double-struck} + and f ∈L 1(ω)) for some φ∈L ∞(1/ω). Also, every D φ has a unique extension to a continuous derivation D̄φ:M(ω)→L∞(1/ω) from the corresponding measure algebra. We show that a certain condition on φ implies that D̄ φ is weak-star continuous. The condition holds for instance if φ∈L 0 ∞(1/ω). We also provide examples of functions φ for which D̄ φ is not weak-star continuous. Similarly, we show that D φ and D̄φ are compact under certain conditions on φ. For instance this holds if φ∈C 0(1/ω) with φ(0)=0. Finally, we give various examples of functions φ for which D φ and D̄ φ are not compact.

AB - Let ω be a continuous weight on R+ and let L 1(ω) be the corresponding convolution algebra. By results of Grønbæk and Bade & Dales the continuous derivations from L 1(ω) to its dual space L ∞(1/ω) are exactly the maps of the form (D φf)(t)=∫ 0 ∞f(s)s/t+sφ(t+s)ds (t∈R{double-struck} + and f ∈L 1(ω)) for some φ∈L ∞(1/ω). Also, every D φ has a unique extension to a continuous derivation D̄φ:M(ω)→L∞(1/ω) from the corresponding measure algebra. We show that a certain condition on φ implies that D̄ φ is weak-star continuous. The condition holds for instance if φ∈L 0 ∞(1/ω). We also provide examples of functions φ for which D̄ φ is not weak-star continuous. Similarly, we show that D φ and D̄φ are compact under certain conditions on φ. For instance this holds if φ∈C 0(1/ω) with φ(0)=0. Finally, we give various examples of functions φ for which D φ and D̄ φ are not compact.

U2 - 10.1016/j.jmaa.2012.07.057

DO - 10.1016/j.jmaa.2012.07.057

M3 - Journal article

SN - 0022-247X

VL - 397

SP - 402

EP - 414

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -