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Cauchy–Schwarz inequalities for inner product type transformers in \(\hbox {Q}^*\) norm ideals of compact operators

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Let \(\int _\Omega {\vert {\!\!\;\vert {\,\!A_t h\!\,}\vert \!\!\;}\vert ^2 +\vert {\!\!\;\vert {\,\!B_t^* h\!\,}\vert \!\!\;}\vert ^2}\,d\mu (t)<{+\infty }\) for all h in a Hilbert space \({\mathcal {H}},\) for some \(\hbox {weakly}^*\)-measurable families \(\{A_t\}_{t\in \Omega }\) and \( \{B_t\}_{t\in \Omega }\) of bounded operators on \({\mathcal {H}},\) where at least one of them consists of mutually commuting normal operators. If \(p\geqslant 2, \Phi \) is a symmetrically norming (s.n.) function, \({\Phi ^{^(\;\!\!^{p}\;\!\!^)}}\) is its p-modification, \({\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\) is a s.n. function adjoint to \({\Phi ^{^(\;\!\!^{p}\;\!\!^)}}\) and \(\vert {\!\!\;\vert {\,\!\cdot \!\,}\vert \!\!\;}\vert _{{\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}}\) is a norm on the ideal associated to s.n. function \({\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\!,\) then for all

$$\begin{aligned} \biggl \vert {\!\!\;\biggl \vert {\,\!\int _\Omega A_t X B_t\,d\mu (t)\,\!}\biggr \vert \!\!\;}\biggr \vert _{{\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}} \leqslant \biggl \vert {\!\!\;\biggl \vert {\,\!\biggl ({\int _\Omega A_t^* A_t\,d\mu (t)}\biggr )^{1/2}\! X \, \biggl ({\int _\Omega B_t B_t^*\,d\mu (t)}\biggr )^{1/2}\,\!}\biggr \vert \!\!\;}\biggr \vert _{{\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}}. \end{aligned}$$

This enable us to prove that if \(\mu \) is a complex Borel measure on \({{\mathbb {R}}}_+,\) with its total variation \(|\mu |({{\mathbb {R}}}_+)\leqslant 1\) and are such that AB are dissipative and at least one of them is normal, such that then

$$\begin{aligned}&\bigl \vert {\!\!\;\bigl \vert {\,\!\!\sqrt{iA^*-iA}\, \bigl ({{\hat{\mu }}(A)X-X{\hat{\mu }}(B)}\bigr )\!\sqrt{iB^*-iB}\,\!}\bigr \vert \!\!\;}\bigr \vert _{\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\\&\quad \leqslant \biggl \vert {\!\!\;\biggl \vert {\,\!\!\sqrt{I-\bigl \vert {{\hat{\mu }}(A)}\bigr \vert ^2}({AX-XB}) \!\sqrt{I-\bigl \vert {{\hat{\mu }}(B)^*}\bigr \vert ^2}\,\!}\biggr \vert \!\!\;}\biggr \vert _{\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\!. \end{aligned}$$

Some others norm inequalities for operator valued and transformer valued Fourier transformations are also provided.

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Correspondence to Milan Lazarević.

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All authors were partially supported by MPNTR Grant No. 174017, Serbia.

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Jocić, D.R., Krtinić, Ɖ. & Lazarević, M. Cauchy–Schwarz inequalities for inner product type transformers in \(\hbox {Q}^*\) norm ideals of compact operators. Positivity 24, 933–956 (2020). https://doi.org/10.1007/s11117-019-00710-3

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