Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

Question

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES.

And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, namely, for any tow matrices $A,B \in \mathbb{R}^{m\times T}$, $\forall \ 0<p\le1$, we have

$$ \lVert A+B \rVert_{S_p}^p \le \lVert A \rVert_{S_p}^p + \lVert B \rVert_{S_p}^p $$

The$\lVert \cdot \rVert_{S_p}$means the Schatten-p norm, $$ \lVert A \rVert_{S_p}=\left(\sum_{j=1}^{\min\{m,T\}}\sigma_j(A)^p\right)^{1/p} $$ And $\sigma_j(A)$ means the singular value of $A$.

The authors say that we could find the result from MCCARTHY, C. A. (1967). $C_p$ or ROTFELD, S. Y. (1969). The singular numbers of the sum of completely continuous operators. But both papers are about the some compact operators on $\textbf{a}$ Hilbert space, from my perspective, the results from the both papers above can show the inequality is right when $A,B$ are square matrices, but how can it hold for the rectangular matrices?

I'm very puzzled, and hope for the answer sincerely.

THANK YOU!

Answer 1

One can deduce the non-rectangular case from the rectangular one as follows. Let $$P\colon \mathbb{R}^{m+T}\to \mathbb{R}^T,\, \iota\colon \mathbb{R}^m\to \mathbb{R}^{m+T}$$ be the obvious orthogonal projection and the isometric imbedding respectively. Then for any linear operator $F\colon \mathbb{R}^T\to \mathbb{R}^m$ one has equality of the Shatten norms $$||F||_{S_p}=||\iota\circ F\circ P||_{S_p}.$$ Hence replace your operators $A$ and $B$ with their compositions with $\iota$ and $P$ and apply to the compositions the result from the references.