= Support of a positive operator
{title2=$\operatorname{supp}A=(\ker A)^\perp$}
The <support of a positive operator> $A$ is the closed subspace
$$
\operatorname{supp}A=(\ker A)^\perp=\overline{\operatorname{ran}A}.
$$
The range identity follows from <image-kernel orthogonality for an adjoint> and $A=A^*$. Equivalently, it is the range of the <spectral projection> $E_A((0,\infty))$. In finite dimension, it is the <span> of eigenvectors with positive <eigenvalues>, and the range is already closed.
Closure matters in infinite dimension. On the <l2 sequence space>, $Ae_n=e_n/n$ has zero kernel and hence full support, but is not onto: $(1/n)$ is square-summable and its formal inverse image $(1,1,\ldots)$ is not. In <quantum relative entropy>, containment of the first state's support in the reference state's support is necessary for finiteness; finite dimensions also make it sufficient, while infinite dimensions can have divergence despite that containment.
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