Every $C_0$-Semigroup on $L^\infty$ admits a bounded Generator

Question

I remember my professor telling us a statemen like:

Every $C_0$-Semigroup on $L^\infty$ admits a bounded Generator.

I currently need a result o this form, so i wonder if this ture. I couldn't find anything about this neither on google nor in my books. Do you remember a statement similiar to this?

Look up the Lindblad generator I think. – Paul Oct 04 '17 at 09:30 — Paul, Oct 04 '17 at 09:30

score 1 · Accepted Answer · answered Oct 06 '17 at 03:25

As proved by Heinrich P. Lotz, every strongly continuous semigroup of operators on a Grothendieck space with the Dunford-Pettis property is indeed uniformly continuous.

As $L^\infty$ is a Grothendieck space with the Dunford-Pettis property, the result is true.

The relevant definitions, the proof and examples of other spaces in this class can be found in this book.

Every $C_0$-Semigroup on $L^\infty$ admits a bounded Generator

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