Laplace transformers in norm ideals of compact operators

Let $$p\geqslant 2, \Phi$$ a symmetrically norming (s.n.) function (resp. $${\Phi ^{^(\;\!\!^{p}\;\!\!^)}}$$ its p-modification and $${\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}$$ a s.n. function adjoint to $${\Phi ^{^(\;\!\!^{p}\;\!\!^)}}$$ ) and $${{\varvec{\mathcal {C}}}}_{\;\!\!\Phi }({\mathcal {H}})$$ (resp. $${{\varvec{\mathcal {C}}}}_{\;\!\!\Phi ^{{^(\;\!\!^{p}\;\!\!^)}\;\!\!}}\;\!\!({\mathcal {H}})$$ and $${{\varvec{\mathcal {C}}}}_{\;\!\!\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\!({\mathcal {H}})$$ ) be its associated ideals of compact operators acting on a Hilbert space $${\mathcal {H}}$$ and let $$f,g,h:[0,{+\infty })\rightarrow {{\mathbb {C}}}$$ be Lebesgue measurable functions. Some recently obtained Cauchy–Schwarz-type norm inequalities were used to systematically explore a class of Laplace transformers of the form $${\mathcal {L}}f({{\Delta _{A,B}}}) :X\mapsto \int _{[0,{+\infty })} e^{-tA}$$ $$Xe^{-tB} f(t)\,{\rm d}t \,\bigl ({=\int _{[0,{+\infty })} e^{-t{\Delta _{A,B}}}X f(t)\,{\rm d}t}\bigr ),$$ acting on the $${{\varvec{\mathcal {B}}}}({\mathcal {H}}),$$ $${{\varvec{\mathcal {C}}}}_{\;\!\!\Phi }({\mathcal {H}}) ,$$ $${{\varvec{\mathcal {C}}}}_{\;\!\!\Phi ^{{^(\;\!\!^{p}\;\!\!^)}\;\!\!}}\;\!\!({\mathcal {H}})$$ or $${{\varvec{\mathcal {C}}}}_{\;\!\!\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\!({\mathcal {H}}) ,$$ induced by a generalized derivation $${\Delta _{A,B}}:{{\varvec{\mathcal {B}}}}({\mathcal {H}})$$ $$\rightarrow {{\varvec{\mathcal {B}}}}({\mathcal {H}}):X\mapsto AX+XB$$ and $${{\varvec{\mathcal {B}}}}({{\varvec{\mathcal {B}}}}({\mathcal {H}}))$$ valued Laplace transform $${\mathcal {L}}f$$ of a function f. If $$\int _{\,\![0,{+\infty })}\!({\vert {\!\,\!\vert {\,\!e^{-tA}x\!\,}\vert \!\,\!}\vert ^2|f(t)|^2+\vert {\!\,\!\vert {\,\!e^{-tB}x\!\,}\vert \!\,\!}\vert ^2|g(t)|^2}) \,{\rm d}t<\!\,\!{+\infty }$$ for all $$x\in {\mathcal {H}}$$ and both A and B are normal, then for all $$X\in {{\varvec{\mathcal {C}}}}_{\;\!\!\Phi }({\mathcal {H}})$$ $$\begin{aligned}&\biggl \vert {\!\biggl \vert {\,\!\int _{[0,{+\infty })} e^{-tA}Xe^{-tB} f(t)g(t)\,{\rm d}t\,\!}\biggr \vert \!}\biggr \vert _\Phi \\&\quad \leqslant \biggl \vert {\!\biggl \vert {\,\!\biggl ({\int _{[0,{+\infty })} e^{-t(A^*+A)} |f(t)|^2\,{\rm d}t}\biggr )^{\!\!\,\!\frac{1}{2}}\! X \biggl ({\int _{[0,{+\infty })} e^{-t(B^*+B)} |g(t)|^2\,{\rm d}t}\biggr )^{\!\!\,\!\frac{1}{2}}\,\!}\biggr \vert \!}\biggr \vert _\Phi . \end{aligned}$$ If $$\alpha ,\beta \in [0,1],$$ then for all $$X\in {{\varvec{\mathcal {C}}}}_{\;\!\!\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\!({\mathcal {H}})$$ $$\begin{aligned}&\vert {\!\,\!\vert {\,\!{\mathcal {L}}(f*g)({\Delta _{A,B}})X\!\,}\vert \!\,\!}\vert _{\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\!\;\!\!\!\,\!\\&\quad \leqslant \bigl \vert {\!\bigl \vert {\,\!\!\,\!\bigl ({{\mathcal {L}}\bigl ({|f|^{2-2\alpha }\!\;\!\!*\;\!\!|g|^{2-2\beta }}\bigr )( {\Delta _{A^*\!,A}})(I)\!\,\!}\bigr )^{\!\frac{1}{2}} X\bigl ({{\mathcal {L}}\bigl ({|f|^{2\alpha }\!\;\!\!*\;\!\!|g|^{2\beta }}\bigr )( {\Delta _{B\!\,\!,B^*\!}})(I)\!\,\!}\bigr )^{\!\frac{1}{2}}\!\,\!}\bigr \vert \!}\bigr \vert _{\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\!, \end{aligned}$$ whenever $$\int \limits _{[0,{+\infty })}\!\!\!\,\!({\vert {\!\,\!\vert {\,\!e^{-tA}\;\!\!x\!\,}\vert \!\,\!}\vert ^2{|f|^{2-2\alpha }\!*\!\,\!|g|^{2-2\beta }(t)} \!\,\!+\;\!\!\vert {\!\,\!\vert {\,\!e^{-tB^*}\!\!x\!\,}\vert \!\,\!}\vert ^2{|f|^{2\alpha }\!*\!\,\!|g|^{2\beta }(t)}\!\,\!})\;\!{\rm d}t$$ $$<{+\infty }$$ for all $$x\in {\mathcal {H}}$$ and at least one of operators A or B is normal, where $${\mathcal {L}}h(C)\, {\mathop =\limits ^{\text {\tiny def}}}\,\int _{[0,{+\infty })} e^{-tC} h(t)\,{\rm d}t$$ denotes the operator valued Laplace transform of a function h and $$f*g$$ denotes a convolution function $$f*g(t)\, {\mathop =\limits ^{\text {\tiny def}}}\,\int _{[0,t]}f(t-s)g(s)\;\!\mathrm{d}s$$ for all $$t\geqslant 0.$$ Applications of Laplace transformers to norm inequalities include the norm inequality $$\begin{aligned}&\bigl \vert {\!\bigl \vert {\,\!({A^{*2}+2A^*A+A^2})^{\!\,\!1/2}X({B^2+2BB^*+B^{*2}})^{\!\,\!1/2}\;\!\!\,\!}\bigr \vert \!}\bigr \vert _{\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\\&\quad \leqslant \bigl \vert {\!\bigl \vert {\,\!A^2X+2AXB+XB^2\,\!}\bigr \vert \!}\bigr \vert _{\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}, \end{aligned}$$ if $$A,B,X\in {{\varvec{\mathcal {B}}}}({\mathcal {H}})$$ are such that $$A,B^*$$ are 2-hyper-accretive and at least one of them is normal, satisfying $$A^2X+2AXB+XB^2\in {{\varvec{\mathcal {C}}}}_{\;\!\!\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\!({\mathcal {H}}) .$$