Support of a positive operator ($\mathrm{supp}A=(\ker A{)}^{\perp }$)

The support of a positive operator $A$ is the closed subspaceThe range identity follows from image-kernel orthogonality for an adjoint and $A=A^{\ast }$. 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, $A{e}_n=e_n/n$ has zero kernel and hence full support, but is not onto: $(1/n)$ is square-summable and its formal inverse image $(1,1,\dots )$ 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.