$\int \lambda d E_\lambda^A \cdot \int( 1/\lambda) \chi_{(0,\infty)} d E_\lambda^A =?$

Question

Let $A$ be a positive operator acting on a Hilbert space $H$. Let $$ A=\int \lambda d E_\lambda^A$$ be the spectral decomposition. Define $B:=\int (1/\lambda) \chi_{(0,\infty)} d E_\lambda ^A$. Then, clearly, $B$ can be unbounded. I would like to ask do we have $AB = s(A)$ (the support of $A$)? or do we have $AB\le s(A)$.