Monotonicity of $p$-norms of multiple operators via unitary swivels

Abstract

Following the various statements of [DW16] to their logical conclusion, this note explicitly argues the following statement, implicit in [DW16]: for positive semi-definite operators $C_1,\dots ,C_L$, a unitary $V_{C_i}$ commuting with $C_i$, and $p\ge 1$, the quantity $$\underset{V_{C_1},\dots ,V_{C_L}}{\max }{\Vert C_1^{1/p}{V}_{C_1}\cdots C_L^{1/p}{V}_{C_L}\Vert }_p^p$$ is monotone non-increasing with respect to $p$. The idea from [DW16] is that by allowing unitary swivels connecting a long chain of positive semi-definite operators together, we can establish such a statement, which might not hold generally without the presence of the unitary swivels. Other related statements follow directly from [DW16] as well, being implicit there, and are given explicitly in this note.

5 pages

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