### Abstract

We revisit and prove some convexity inequalities for trace functions conjectured in this paper's antecedent. The main functional considered is Φ_{{p,q}} (A_{1}, A_{2}, ..., A_{m}) = (Tr[(∑_{j=1}^{m} A_{j}^{p}) ^{q/p}])^{1/q} for m positive definite operators A _{j} . In our earlier paper, we only considered the case q = 1 and proved the concavity of Φ _{p,1} for 0 < p ≤ 1 and the convexity for p = 2. We conjectured the convexity of Φ _{p,1} for 1 < p < 2. Here we not only settle the unresolved case of joint convexity for 1 ≤ p ≤ 2, we are also able to include the parameter q ≥ 1 and still retain the convexity. Among other things this leads to a definition of an L ^{q} (L ^{p} ) norm for operators when 1 ≤ p ≤ 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces - which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.

Original language | English (US) |
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Pages (from-to) | 107-126 |

Number of pages | 20 |

Journal | Letters in Mathematical Physics |

Volume | 83 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2008 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

- Concavity
- Convexity
- Entropy
- Operator norms
- Trace inequality

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## Cite this

*Letters in Mathematical Physics*,

*83*(2), 107-126. https://doi.org/10.1007/s11005-008-0223-1