A pinching over $M_n({\mathbb C})$ is an endomorphism $T$ where the $(i,j)$-entry of $T(M)$ is given either by $0$ or by $m_{ij}$, depending on the pair $(i,j)$. Let us say that a pinching is symmetric if the rule is the same for $(i,j)$ and $(j,i)$ whenever $j\ne i$.
R. Bhatia has shown that the pinching $M\mapsto D(M):={\rm diag}(m_{11},\ldots,m_{nn})$ is a contraction for every unitarily invariant norm $\|\cdot\|$. In other words,
$$\|D(M)\|\le\|M\|,\qquad\forall M\in M_n({\mathbb C}).$$
Remark that the map $D$ sends the cone of positive definite Hermitian matrices $HPD_n$ into itself.
It is not difficult to extend Bhatia's result to block pinching $\Delta$, in which $\Delta(M)$ is block diagonal, made from diagonal blocks of $M$. Again $\Delta$ sends $HPD_n$ into itself. On an other hand, it is known that some non-block diagonal pinching are not contracting and do not preserve $HPD_n$. For instance, the linear map
$$M=\left( \begin{array}{ccc}
a & b & c \\\\
d & e & f \\\\ g & h & k \end{array} \right)\mapsto B(M)=\left( \begin{array}{ccc}
a & b & 0 \\\\
d & e & f \\\\ 0 & h & k \end{array} \right).$$
The operator norm of $B$ (when $M_n({\mathbb C})$ is endowed with a unitarily invariant norm) is larger than $1$ (Bhatia), and there exists $H\in HPD_n$ such that $B(M)$ is even not semi-positive definite.
Question. Are the following three properties equivalent ?
- The pinching $T$ is symmetric and contracting for every unitarily invariant norm of $M_n({\mathbb C})$.
- The pinching $T$ sends $HPD_n$ into itself.
- The pinching $T$ is block diagonal.
Best Answer
R. Bhatia proved (in Amer. Math. Monthly 107) that the operator $D_k$ taking $M$ to its $k$-th diagonal ($M_{ij}$, $j-i=k$) contracts any unitarily invariant norm, so 1 implies neither of 2, 3. Still, it is an interesting question to characterize those "contractive pinchings".
EDIT (partly answering the modified question): Consider a symmetric pinching, $M\mapsto P(M)=(p_{ij}M_{ij})_{ij}$, $p_{ij}=p_{ji}\in\{O,1\}$. Property (2) implies $p_{ii}=1$ for all $i$, since a positive definite matrix has positive diagonal elements. Property (1) doesn't imply the same thing, since all $p_{ij}$ might be zero (as by Guillaume's comment) [EDIT: a slightly less trivial example is $p_{11}=p_{22}=0$, $p_{12}=p_{21}=1$, and $p_{ij}=\delta_{i,j}$ if $i>2$ or $j>2$ ]. So let us assume $p_{ii}=1$ as part of the hypothesis. Then both (1) and (2) imply that the relation between indices $\{(i,j): p_{ij}=1\}$ is reflexive and symmetric. If it is not transitive (i.e. if it is not an equivalence relation, or equivalently (3) doesn't hold), then there are three distinct indices $i,j,k$ with $p_{ij}=p_{jk}=1$ but $p_{ik}=0$. Let $I=\{i,j,k\}$. Then, considering your $B$ example, $P$ doesn't preserve positive definiteness, since the $I \times I$ principal minor of $P(M)$ can be negative for a positive definite $M$. Hence (2) implies (3), and they are equivalent. Similiarly, restricting $P$ to matrices supported on $I\times I$, if $B$ doesn't contract some "natural" unitarily invariant norm on $3\times 3$ matrices,the same must be true of $P$. But it is easily seen that $B$ doesn't contract the trace norm (sum of singular values), since the all ones $3\times3$ matrix $E$ has trace norm $3$ and $B(E)$ has trace norm $1+2\sqrt{2}$ (cf Bhatia). Hence (1),(2),(3) are equivalent, if in (1) one assumes $p_{ii}=1$ for all $i$ (i.e. the symmetric pinching also preserves the diagonal).