On the equality of operator valued weights

G. K. Pedersen and M. Takesaki have proved in 1973 that if $\phi$ is a faithful, semi-finite, normal weight on a von Neumann algebra $M\text{\hspace{0.17em}⁣}$, and $\psi$ is a ${\sigma }^{\phi }$-invariant, semi-finite, normal weight on $M\text{\hspace{0.17em}⁣}$, equal to $\phi$ on the positive part of a weak${}^{\ast }$-dense ${\sigma }^{\phi }$-invariant $\ast$-subalgebra of ${\mathfrak{M}}_{\phi }\text{\hspace{0.17em}⁣}$, then $\psi =\phi \text{\hspace{0.17em}⁣}$. In 1978 L. Zsid'o extended the above result by proving: if $\phi$ is as above, $a\ge 0$ belongs to the centralizer $M^{\phi }$ of $\phi \text{\hspace{0.17em}⁣}$, and $\psi$ is a ${\sigma }^{\phi }$-invariant, semi-finite, normal weight on $M\text{\hspace{0.17em}⁣}$, equal to ${\phi }_a:=\phi (a^{1\mathrm{/}2}\text{\hspace{0.17em}⁣}\cdot \text{\hspace{0.17em}⁣}{a}^{1\mathrm{/}2})$ on the positive part of a weak${}^{\ast }$-dense ${\sigma }^{\phi }$-invariant $\ast$-subalgebra of ${\mathfrak{M}}_{\phi }\text{\hspace{0.17em}⁣}$, then $\psi ={\phi }_a\text{\hspace{0.17em}⁣}$. Here we will further extend this latter result, proving criteria for both the inequality $\psi \le {\phi }_a$ and the equality $\psi ={\phi }_a\text{\hspace{0.17em}⁣}$. Particular attention is accorded to criteria with no commutation assumption between $\phi$ and $\psi \text{\hspace{0.17em}⁣}$, in order to be used to prove inequality and equality criteria for operator valued weights. Concerning operator valued weights, it is proved that if $E_1\text{\hspace{0.17em}⁣},E_2$ are semi-finite, normal operator valued weights from a von Neumann algebra $M$ to a von Neumann subalgebra $N\ni 1_M$ and they are equal on ${\mathfrak{M}}_{E_1}\text{\hspace{0.17em}⁣}$, then $E_2\le E_1\text{\hspace{0.17em}⁣}$. Moreover, it is shown that this happens if and only if for any (or, if $E_1\text{\hspace{0.17em}⁣},E_2$ have equal supports, for some) faithful, semi-finite, normal weight $\theta$ on $N$ the weights $\theta \circ E_2\text{\hspace{0.17em}⁣},\theta \circ E_1$ coincide on $\mathfrak{M}_{\theta\circ E_1};!$.