# About the closedness of a certain product of two closed operators?

Let $$H$$ be a Hilbert space. Let $$S∈B(H)$$ and let $$T$$ be a densely defined closed operator such that $$TS\subset ST$$. Assume further that $$T$$ is boundedly invertible.

Is it true that $$ST$$ is closed?

In my problem, $$S$$ and $$T$$ are both self adjoint and positive.

• What does boundedly invertible mean? $T: D(T) \to H$ is injective and $T^{-1}: T(D(T)) \to H$ is bounded? Jan 15 at 20:22
• Are you also sure that you don't mean $ST \subseteq TS$? Jan 15 at 20:30
• Yes. It is not the usual commutativity... Jan 15 at 20:34
• Any ideas or suggestions? Jan 16 at 12:22