Refinements of operator Cauchy-Schwarz and Minkowski inequalities for p-modified norms and related norm inequalities

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For (degree) p -modifications Φ^(p)

Φ^{(p)}

of symmetrically norming functions Φ and finite families of bounded Hilbert space operators A₁,B₁,⋯,A_N,B_N

A_{1}, B_{1}, \dots, A_{N}, B_{N}

and X

\begin{matrix} {‖ \sum_{n = 1}^{N} A_{n}^{⁎} X B_{n} ‖}_{Φ^{(p)}} \\ \leq ‖ {(| \sum_{n = 1}^{N} A_{n}^{⁎} X B_{n} |}^{2} + \sum_{n = 1}^{N} | A_{n} {(‖ \sum_{n = 1}^{N} A_{n}^{⁎} A_{n} ‖}^{\frac{1}{2}} I \\ {+ {(‖ \sum_{n = 1}^{N} A_{n}^{⁎} A_{n} ‖ I - \sum_{n = 1}^{N} A_{n}^{⁎} A_{n})}^{\frac{1}{2}})}^{- 1} \sum_{m = 1}^{N} A_{m}^{⁎} X B_{m} \\ {{{{- ‖ \sum_{n = 1}^{N} A_{n}^{⁎} A_{n} ‖}^{\frac{1}{2}} X B_{n} |}^{2})}^{\frac{p}{2}} ‖}_{Φ}^{\frac{1}{p}} \\ {{= ‖ \sum_{n = 1}^{N} A_{n}^{⁎} A_{n} ‖}^{\frac{1}{2}} ‖ {(\sum_{n = 1}^{N} B_{n}^{⁎} X^{⁎} X B_{n})}^{\frac{1}{2}} ‖}_{Φ^{(p)}} \end{matrix}

for 0<p≤2

0 < p \leq 2

, while for p≥2

p \geq 2

\begin{matrix} {‖ \sum_{n = 1}^{N} A_{n}^{⁎} X B_{n} ‖}_{Φ^{(p)}} \\ {\leq ‖ | \sum_{n = 1}^{N} A_{n}^{⁎} X B_{n} |}^{p} + \sum_{n = 1}^{N} | A_{n} {(‖ \sum_{n = 1}^{N} A_{n}^{⁎} A_{n} ‖}^{\frac{1}{2}} I \\ {+ {(‖ \sum_{n = 1}^{N} A_{n}^{⁎} A_{n} ‖ I - \sum_{n = 1}^{N} A_{n}^{⁎} A_{n})}^{\frac{1}{2}})}^{- 1} \sum_{m = 1}^{N} A_{m}^{⁎} X B_{m} \\ {{{- ‖ \sum_{n = 1}^{N} A_{n}^{⁎} A_{n} ‖}^{\frac{1}{2}} X B_{n} |}^{p} ‖}_{Φ}^{\frac{1}{p}} \\ {{\leq ‖ \sum_{n = 1}^{N} A_{n}^{⁎} A_{n} ‖}^{\frac{1}{2}} ‖ \sum_{n = 1}^{N} B_{n}^{⁎} X^{⁎} X B_{n} ‖}_{Φ^{(\frac{p}{2})}} . \end{matrix}

For p≥2 $p \geq 2$ this implies refined Minkowski inequality for p -modified unitarily invariant norms ‖⋅‖_Φ^(p) ${‖ \cdot ‖}_{Φ^{(p)}}$

\begin{matrix} {{‖ \sum_{n = 1}^{N} B_{n} ‖}_{Φ^{(p)}} \leq ‖ | \sum_{n = 1}^{N} B_{n} |}^{p} \\ + {(\sum_{n = 1}^{N} {‖ B_{n} ‖}_{Φ^{(p)}})}^{- \frac{p}{2}} \sum_{n = 1}^{N} {‖ B_{n} ‖}_{Φ^{(p)}}^{- \frac{p}{2}} | {‖ B_{n} ‖}_{Φ^{(p)}} \sum_{m = 1}^{N} B_{m} \\ {{- \sum_{m = 1}^{N} {‖ B_{m} ‖}_{Φ^{(p)}} B_{n} |}^{p} ‖}_{Φ}^{\frac{1}{p}} \leq \sum_{n = 1}^{N} {‖ B_{n} ‖}_{Φ^{(p)}} . \end{matrix}

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