2 fixed missing square-root

Some results in this direction that you might find useful are listed below.

Theorem (Bhatia and Kittaneh). Let $X$, $Y$, and $Z$ be as in the question above. Then, \begin{equation*} \| (X^2+Y^2)^{1/2} \|_p \le \|Z\|_p \le 2^{1/2-1/p}\| (X^2+Y^2)^{1/2} \|_p, \end{equation*} where $2 \le p \le \infty$, and $\|\cdot\|_p$ denotes the Schatten-$p$ norm. The inequality above gets reversed for $1\le p \le 2$. Also, these inequalities are sharp.

Even more directly relevant is the following theorem that discusses majorization of singular values of $X+Y$ by those of $Z$.

Theorem (Bhatia and Kittaneh). Let $X$, $Y$, and $Z$ be as in the question. Then \begin{equation*} \sigma(X+Y)\quad \prec_w\quad \sqrt{2}\sigma(Z) \end{equation*} If $X$ is psd, then the above weak majorization can be replaced by weak log-majorization, that is, \begin{equation*} \prod_{j=1}^k \sigma_j(X+Y) \le \prod_{j=1}^k\sqrt{2}\sigma_j(Z). \end{equation*} Finally, if both $X$ and $Y$ are psd, then we have the even stronger inequalities: \begin{equation*} \sigma_j(X+Y) \le \sigma_j(Z)\quad sqrt{2}\sigma_j(Z)\quad 1 \le j \le n. \end{equation*}

Bhatia and Kittaneh also discuss some applications of the above theorem to commutator inequalities.

References

1. R. Bhatia and F. Kittaneh. "The singular values of $A+B$ and $A+iB$." Linear Algebra and its Applications, 431(2009), pp. 1502-1508.

2. R. Bhatia and F. Kittaneh. "Cartesian decompositions and Schatten norms." Linear Algebra and its Applications, 318(2000), pp. 109--116.

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Some results in this direction that you might find useful are listed below.

Theorem (Bhatia and Kittaneh). Let $X$, $Y$, and $Z$ be as in the question above. Then, \begin{equation*} \| (X^2+Y^2)^{1/2} \|_p \le \|Z\|_p \le 2^{1/2-1/p}\| (X^2+Y^2)^{1/2} \|_p, \end{equation*} where $2 \le p \le \infty$, and $\|\cdot\|_p$ denotes the Schatten-$p$ norm. The inequality above gets reversed for $1\le p \le 2$. Also, these inequalities are sharp.

Even more directly relevant is the following theorem that discusses majorization of singular values of $X+Y$ by those of $Z$.

Theorem (Bhatia and Kittaneh). Let $X$, $Y$, and $Z$ be as in the question. Then \begin{equation*} \sigma(X+Y)\quad \prec_w\quad \sqrt{2}\sigma(Z) \end{equation*} If $X$ is psd, then the above weak majorization can be replaced by weak log-majorization, that is, \begin{equation*} \prod_{j=1}^k \sigma_j(X+Y) \le \prod_{j=1}^k\sqrt{2}\sigma_j(Z). \end{equation*} Finally, if both $X$ and $Y$ are psd, then we have the even stronger inequalities \begin{equation*} \sigma_j(X+Y) \le \sigma_j(Z)\quad 1 \le j \le n. \end{equation*}

Bhatia and Kittaneh also discuss some applications of the above theorem to commutator inequalities.

References

1. R. Bhatia and F. Kittaneh. "The singular values of $A+B$ and $A+iB$." Linear Algebra and its Applications, 431(2009), pp. 1502-1508.

2. R. Bhatia and F. Kittaneh. "Cartesian decompositions and Schatten norms." Linear Algebra and its Applications, 318(2000), pp. 109--116.