For every integer N≥ 1, every finite-dimensional Hilbert spaces E_1,…,E_N,F_1,F_2, every block operator A=[A_ij]_i=1,j=1^2,N with A_ij:E_j→ F_i, and every 1≤ p≤ 2, one has |[|A_ij|_S_p]_i=1,j=1^2,N|_S_p≤ |[A_ij]_i=1,j=1^2,N|_S_p, where the matrix on the left is the scalar 2× N matrix of block Schatten p-norms.

Lin–van den Driessche conjecture on Schatten norm compression

For every integer $N\ge 1$ , every finite-dimensional Hilbert spaces $E_1,\ldots,E_N,F_1,F_2$ , every block operator $A=[A_{ij}]_{i=1,j=1}^{2,N}$ with $A_{ij}:E_j\to F_i$ , and every $1\le p\le 2$ , one has $\left\|[\|A_{ij}\|_{S_p}]_{i=1,j=1}^{2,N}\right\|_{S_p}\le \left\|[A_{ij}]_{i=1,j=1}^{2,N}\right\|_{S_p}$ , where the matrix on the left is the scalar $2\times N$ matrix of block Schatten $p$ -norms.

References

Primary source

arXiv

Additional references

Three-Point Trace Positivity:Schatten Norm Compression and Interpolating Metrics — arXiv — Trung Dung Vuong, Hiroyuki Osaka

Posted 3 hours ago · edited 2 hours ago

Progress summary

Refreshed 2 hours ago

Open

The conjecture remains open, with only special cases established in the retrieved evidence.

The conjecture concerns whether replacing blocks of a partitioned $2\times N$ matrix by their Schatten $p$ -norms decreases the resulting norm for $1\le p\le2$ . The retrieved literature records partial results but does not establish the full statement.

Known results

The conjectured inequality is reported for several special cases, including Hanner’s matrix inequality as a special case.
For $3\times3$ and larger block partitions, examples show that the proposed inequality can fail.
Some special cases extend to $0\le p\le1$ , but this extension is not valid in general.

Current status (as of October 2026): The general Schatten compression conjecture for $1\le p\le2$ remains open; the retrieved evidence establishes only special cases.

Sources

No solutions have been posted yet.