Noncommutative Pick–Julia theorems for generalized derivations in Q, Q\(^*\) and Schatten–von Neumann ideals of compact operators

Abstract

If C and D are strictly accretive operators on \({\mathcal {H}}\) and at least one of them is normal, such that \(CX\!-\!XD\in { {{{\varvec{{\mathcal {C}}}}}}_{\Psi }({\mathcal {H}})}\) for some \(X\in { {{{\varvec{{\mathcal {B}}}}}}({\mathcal H})}\) and \(Q^*\) symmetrically norming function \(\Psi ,\) then for all holomorphic functions h,  mapping the open right half (complex) plane into itself, we have \(h( C\,\!)X\!-\!Xh(D)\in { {{{\varvec{{\mathcal {C}}}}}}_{\Psi }({\mathcal {H}})},\) satisfying

$$\begin{aligned}&\bigl \vert {\,\!\bigl \vert {(C^*\!+C)^{ 1/2}\bigl ({h( C\,\!){}X\!-\!Xh(D)\!\,\!}\bigr )(D+D^*\!\,\!)^{ 1/2}}\bigr \vert \,\!}\bigr \vert _\Psi \\&\leqslant \bigl \vert {\,\!\bigl \vert {\bigl ({h( C\,\!){}^*\!+h( C\,\!){}\!\,\!}\bigr )^{ 1/2}{({ CX\!-\!XD})} \bigl ({h(D)+h(D)^*\!\,\!}\bigr )^{ 1/2}}\bigr \vert \,\!}\bigr \vert _\Psi . \end{aligned}$$

If \(1\leqslant q,r,s\leqslant {+\infty }\) and \(p\geqslant 2,A,B,X\in { {{{\varvec{{\mathcal {B}}}}}}({\mathcal H})}\) and A, B are strict contractions satisfying the condition \(AX\!-\!XB\in { {{{\varvec{{\mathcal {C}}}}}}_{s}({\mathcal {H}})},\) then for all holomorphic functions g,  mapping the open unit disc into the open right half (complex) plane, \(g(A)X\!-\!Xg(B)\in { {{{\varvec{{\mathcal {C}}}}}}_{s}({\mathcal {H}})},\) satisfying Schatten–von Neumann s-norms \((\vert {\;\!\vert {\cdot }\vert \;\!}\vert _s)\) inequality

$$\begin{aligned}&\,\!\Bigl \vert \!\,\!\Bigl \vert {\bigl \vert {\!\,\!\bigl ({g(A)^{*}\!+g(A)\!\,\!}\bigr )^\frac{1}{2}\!{({I\!-\!A^{*}\!A})}^\frac{1}{2}\!\,\!}\bigr \vert ^{\!\frac{1}{q}-1} \!\,\!{({I\!-\!A^{*}\!A})}^\frac{1}{2}\!\bigl ({g(A)X\!-\!Xg(B)\!\,\!}\bigr )}\Bigr .\Bigr .\\&\times \Bigl .\Bigl .{{({I\!-BB^*\!\,\!})}^\frac{1}{2}\!\bigl \vert {\!\,\!\bigl ({g(B)+g(B)^{*}\!\,\!}\bigr )^\frac{1}{2}\!{({I\!-BB^*\!\,\!})}^\frac{1}{2}\!\,\!}\bigr \vert ^{\!\frac{1}{r}-1}\!\,\!}\Bigr \vert \!\,\!\Bigr \vert _s \\ \leqslant&\,\!\Bigl \vert \!\,\!\Bigl \vert {\bigl \vert {\!\,\!\bigl ({g(A)^{*}\!+g(A)\!\,\!}\bigr )^\frac{1}{2}\!{({I\!-\!AA^{*}\!\,\!})}^\frac{1}{2}\!\,\!}\bigr \vert ^\frac{1}{q} {({I-AA^*\!\,\!})}^{\!\,\!-\frac{1}{2}}\!\,\!{({AX\!-\!XB})}}\Bigr .\Bigr .\\&\times \Bigl .\Bigl .{{({I-B^*\!B})}^{\!\,\!-\frac{1}{2}}\!\,\!\bigl \vert {\!\,\!\bigl ({g(B)+g(B)^{*} \!\,\!}\bigr )^\frac{1}{2} \!{({I-B^*\! B})}^\frac{1}{2}\!\,\!}\bigr \vert ^\frac{1}{r}\!\,\!}\Bigr \vert \!\,\!\Bigr \vert _s. \end{aligned}$$

Other variants of some new Pick–Julia-type norm and operator inequalities are also obtained, they both complement the well-known Pick–Julia theorems for operators, obtained by Ky Fan, Ando, and Author, and they also extend these theorems to the field of norm ideals of compact operators, including Schatten–von Neumann ideals.