Norm inequalities for the iterated perturbations of Laplace transformers generated by accretive \(\scriptstyle N\)-tuples of operators in Q and Q* ideals of compact operators

Abstract

Let \(\Phi ,\Psi \) be symmetrically norming (s.n.) functions, and \({{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle A\;\!,B}X\;\!{\mathop {=}\limits ^{\tiny {\text {def}}}}\;\!{{{{\mathscr {L}}}}}\;\X\;\!{\mathop {=}\limits ^{\tiny {\text {def}}}}\;\!\int _{{{\mathbb {R}}}_+}\!e^{\!-tA}Xe^{\!-tB}\;\!d\mu (t)\) denotes the Laplace transformer generated by the generalized derivation where \(\mu \) is a Borel probability measure on If both pairs consist of mutually commuting accretive operators, such that both \(C\;\!-A\) and \(D-B\) are accretive and for some , then \({{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle A^{\;\!*}\!\!,A}^{}\;\!(I)-{{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle C^*\!\!,C}^{}\;\!(I)\;\!\geqslant \;\!0,{{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle B\;\!,B^*}^{}\;\!(I)-{{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle D\;\!,D^*}^{}\;\!(I)\;\!\geqslant \;\!0\) and

$$\begin{aligned}&\;\!\bigl \vert {\bigl \vert {\!\sqrt{C^*\!\;\!+\!C\!-A^*\!\;\!-\!A}\bigl ({{{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle A\;\!,B}X-{{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle C\;\!,D}X}\bigr )\!\sqrt{D\!+\!\;\!D^*\!-\!B-\!B^*}\;\!}\bigr \vert }\bigr \vert _\Psi \\&\leqslant \;\!\Bigl \vert \Bigl \vert {\textstyle \sqrt{{{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle A^{\;\!*}\!\!,A}^{}\;\!(I)-{{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle C^*\!\!,C}^{}\;\!(I)}\;\!({AX\!+\!XB-CX\!-\!XD})}\Bigr .\Bigr .\\&\times \Bigl .\Bigl .{\!\sqrt{{{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle B\;\!,B^*}^{}\;\!(I)-{{{{\mathscr {L}}}}}\;\![\mu \;\!]\Delta _{\scriptscriptstyle D\;\!,D^*}^{}\;\!(I)}}\Bigr \vert \Bigr \vert _\Psi , \end{aligned}$$

holds under any of the following conditions: (a) if (b) if for some \(p\geqslant 2,{ L^{\;\!2}\;\!(\;\!{{{\mathbb {R}}}_+}\;\!,\mu )}\) is separable and at least one of pairs (A, C) or (B, D) consists of normal operators, (c) if both pairs (A, C) and (B, D) consist of normal operators. Above, \({\Phi ^{^(\;\!\!^{p}\;\!\!^)}}\!\) denotes (the degree) p-modified s.n. function \(\Phi \) and \({\Phi ^{{^(\;\!\!^{p}\;\!\!^)}^{_*}}}\!\!\) is the dual s.n. function for \({\Phi ^{^(\;\!\!^{p}\;\!\!^)}}\!.\) Moreover, the aforementioned inequality is generalized to the iterated perturbations of Laplace transformers, and the alternative inequalities are given for Q norms as well. These inequalities also generalize some previously obtained results.