Let $P$ be a positive $n\times n$ matrix over $\mathbb{C}$, let $f:(0,\mathbb{R})\rightarrow\mathbb{R}$ be a continuous function, and define a function on positive definite $n\times n$ matrices as $$ Q\mapsto \mathrm{Tr}(P f(Q)).\tag{$\ast$} $$ This is well defined for positive definite matrices $Q$, but it can be extended to positive semi-definite matrices $Q$ as long as $\mathrm{null}(Q)\subseteq\mathrm{null}(P)$. We can write this extension in terms of the eigenvalues of $Q$ and $P$ as $$ \mathrm{Tr}(Pf(Q)) = \sum_{i=1}^r\sum_{j=1}^n p_j f(q_i) \lvert \langle u_i,v_j\rangle\rvert^2 $$ where the spectral decompositions of $P$ and $Q$ are $$ P = \sum_{i=1}^n p_i v_iv_i^* \qquad\text{and}\qquad Q = \sum_{i=1}^n q_i u_iu_i^* $$ with $q_1,\dots,q_r>0$ and $q_{r+1}=\cdots=q_{n}=0$.
For example, the quantum relative entropy of two positive $n\times n$ matrices $P,Q\in\mathrm{H}_n^+$ can be defined as $$ D(P\lVert Q) = \mathrm{Tr}(P\log P) - \mathrm{Tr}(P\log Q) $$ whenever $\mathrm{null}(Q)\subseteq\mathrm{null}(P)$ and as $D(P\lVert Q)=+\infty$ otherwise.
It seems that the relative entropy is continuous (when it is extended to $Q$ such that $\mathrm{null}\subseteq\mathrm{null}(P)$). My question is:
- How do we know that $Q\mapsto\mathrm{Tr}(P\log(Q))$ is continuous?
- For a fixed $P$, for which other functions $f$ can the function in ($\ast$) be continuously extended to $Q$ satisfying $\mathrm{null}(Q)\subseteq\mathrm{null}(P)$?
- If $f$ can be continuously extended to $0$, then this function is obviously continuous. What about $f(x)=x^r$ for some $r\leq0$? It seems that $Q\mapsto \mathrm{Tr}(PQ^r)$ is continuous for $r\geq-1$, but I'd like to have a proof if it!
See an example of such functions here.