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tweak to punctuation
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Yemon Choi
Yemon Choi
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As you yourself discovered by finding a paper of Audenaert: an upper bound of the form you require is provided by the Powers—StørmerPowers–Størmer inequality:

Theorem (Powers—StørmerPowers–Størmer, 1970, Lemma 4.1; link) Let $S$ and $T$ be positive Hilbert-Schmidt operators on a Hilbert space. Then $\Vert S - T \Vert_2^2 \leq \Vert S^2-T^2\Vert_1$, where $\Vert \quad\Vert_p$ denotes the Schatten $p$-norm.

The starting idea of the proof is to work in an ONB with respect to which $R=S-T$ is diagonal (which is possible by the spectral theorem for compact self-adjoint operators). One then observes that, putting $Q=S+T$, we have $S^2-T^2 = (RQ+QR)/2$ and then one exploits the fact that $Q\geq \pm R$.

More general inequalities are known, see the discussion in Section X.1 of Bhatia's Springer GTM book on Matrix Analysis (Springer GTM).

As you yourself discovered by finding a paper of Audenaert: an upper bound of the form you require is provided by the Powers—Størmer inequality:

Theorem (Powers—Størmer, 1970, Lemma 4.1; link) Let $S$ and $T$ be positive Hilbert-Schmidt operators on a Hilbert space. Then $\Vert S - T \Vert_2^2 \leq \Vert S^2-T^2\Vert_1$, where $\Vert \quad\Vert_p$ denotes the Schatten $p$-norm.

The starting idea of the proof is to work in an ONB with respect to which $R=S-T$ is diagonal (which is possible by the spectral theorem for compact self-adjoint operators). One then observes that, putting $Q=S+T$, we have $S^2-T^2 = (RQ+QR)/2$ and then one exploits the fact that $Q\geq \pm R$.

More general inequalities are known, see the discussion in Section X.1 of Bhatia's Springer GTM book on Matrix Analysis (Springer GTM).

As you yourself discovered by finding a paper of Audenaert: an upper bound of the form you require is provided by the Powers–Størmer inequality:

Theorem (Powers–Størmer, 1970, Lemma 4.1; link) Let $S$ and $T$ be positive Hilbert-Schmidt operators on a Hilbert space. Then $\Vert S - T \Vert_2^2 \leq \Vert S^2-T^2\Vert_1$, where $\Vert \quad\Vert_p$ denotes the Schatten $p$-norm.

The starting idea of the proof is to work in an ONB with respect to which $R=S-T$ is diagonal (which is possible by the spectral theorem for compact self-adjoint operators). One then observes that, putting $Q=S+T$, we have $S^2-T^2 = (RQ+QR)/2$ and then one exploits the fact that $Q\geq \pm R$.

More general inequalities are known, see the discussion in Section X.1 of Bhatia's Springer GTM book on Matrix Analysis (Springer GTM).

added more details, in particular a link to the original paper of Powers and Stormer
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Yemon Choi
Yemon Choi
  • 25.8k
  • 9
  • 69
  • 156

As you yourself discovered by finding a paper of Audenaert: an upper bound of the form you require is provided by the Powers-StormerPowers—Størmer inequality. See e.g.:

Theorem (Powers—Størmer, 1970, Lemma 4.1; link) Let $S$ and $T$ be positive Hilbert-Schmidt operators on a Hilbert space. Then $\Vert S - T \Vert_2^2 \leq \Vert S^2-T^2\Vert_1$, where $\Vert \quad\Vert_p$ denotes the Schatten $p$-norm.

The starting idea of the proof is to work in an ONB with respect to which A generalization of the Powers-Stormer inequality$R=S-T$ is diagonal (which is possible by the spectral theorem for a statement ofcompact self-adjoint operators). One then observes that, putting $Q=S+T$, we have $S^2-T^2 = (RQ+QR)/2$ and then one exploits the inequalityfact that $Q\geq \pm R$.

I am writingMore general inequalities are known, see the discussion in a rush but will try to come back and supply further references or an outlineSection X.1 of the proofBhatia's Springer GTM book on Matrix Analysis (Springer GTM).

As you yourself discovered by finding a paper of Audenaert: an upper bound of the form you require is provided by the Powers-Stormer inequality. See e.g. A generalization of the Powers-Stormer inequality for a statement of the inequality

I am writing in a rush but will try to come back and supply further references or an outline of the proof.

As you yourself discovered by finding a paper of Audenaert: an upper bound of the form you require is provided by the Powers—Størmer inequality:

Theorem (Powers—Størmer, 1970, Lemma 4.1; link) Let $S$ and $T$ be positive Hilbert-Schmidt operators on a Hilbert space. Then $\Vert S - T \Vert_2^2 \leq \Vert S^2-T^2\Vert_1$, where $\Vert \quad\Vert_p$ denotes the Schatten $p$-norm.

The starting idea of the proof is to work in an ONB with respect to which $R=S-T$ is diagonal (which is possible by the spectral theorem for compact self-adjoint operators). One then observes that, putting $Q=S+T$, we have $S^2-T^2 = (RQ+QR)/2$ and then one exploits the fact that $Q\geq \pm R$.

More general inequalities are known, see the discussion in Section X.1 of Bhatia's Springer GTM book on Matrix Analysis (Springer GTM).

Source Link
Yemon Choi
Yemon Choi
  • 25.8k
  • 9
  • 69
  • 156

As you yourself discovered by finding a paper of Audenaert: an upper bound of the form you require is provided by the Powers-Stormer inequality. See e.g. A generalization of the Powers-Stormer inequality for a statement of the inequality

I am writing in a rush but will try to come back and supply further references or an outline of the proof.

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