Let $B$ be an unbounded closed operator on a Hilbert space $H$. If $B=\int \lambda d E_\lambda $ is positive self-adjoint and a positive bounded operator $X$ commutes with every $E_\lambda $, then why $BX$ is positive and self-adjoint?

I am struggling in dealing with unbounded operators...

see page 48, line +6 (just consider $p=1$) in   [link][1]. I want to understand from line 5 to line 8.

I know it is symmetric, but I have no idea why it is self-adjoint.


  [1]: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7ECF9AE5715F6C8D8A7BA8EC248B28B7/S0305004100058515a.pdf/div-class-title-isometries-of-non-commutative-span-class-italic-l-span-class-sup-p-span-span-spaces-div.pdf