2 results
Matrix inequalities and majorizations around Hermite–Hadamard’s inequality
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 65 / Issue 4 / December 2022
- Published online by Cambridge University Press:
- 10 January 2022, pp. 943-952
- Print publication:
- December 2022
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- Article
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We study the classical Hermite–Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such as the Schatten p-norm estimates
for all positive (semidefinite)
$$ \begin{align*}\left(\|A^q\|_p^p + \|B^q\|_p^p\right)^{1/p} \le \|(xA+(1-x)B)^q\|_p+ \|((1-x)A+xB)^q\|_p, \end{align*} $$
$n\times n$
matrices
$A,B$
and
$0<q,x<1$
. A related decomposition, with the assumption
$X^*X+Y^*Y=XX^*+YY^*=I$
, is
for some family of
$$ \begin{align*}(X^*AX+Y^*BY)\oplus (Y^*AY+X^*BX) =\frac{1}{2n}\sum_{k=1}^{2n} U_k (A\oplus B)U_k^*, \end{align*} $$
$2n\times 2n$
unitary matrices
$U_k$
. This is a majorization which is obtained by using the Hansen–Pedersen trace inequality.
Subadditivity Inequalities for Compact Operators
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 1 / 14 March 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 25-36
- Print publication:
- 14 March 2014
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- Article
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Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional
$\varepsilon $ term. It does not seem possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon $ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also emphasizes matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.
