Cauchy–Schwarz norm inequalities for elementary operators and inner product type transformers generated by families of subnormal operators

Abstract

Recently obtained Cauchy–Schwarz norm inequalities for normal inner product type (i.p.t.) transformers are generalized to i.p.t. transformers generated by some subnormal families of operators. If ${{\mathbf {L}}}^2(\Omega ,\mu )$ is separable, $\Phi$ is a symmetrically norming (s.n.) function and at least one of square integrable families $\{{{A}_t}\}_{t\in \Omega }$ or $\{{{B}_t^*}\}_{t\in \Omega }$ is uniformly extendable to (strongly) square integrable families of commuting normal operators, then for all ${p \geqslant 2}$ and $X\in {{\mathcal {C}}_{\Phi ^{(p)^*}}({\mathcal H})}$

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

If $f(z)=\sum _{n=0}^{\infty }c_nz^n$ is analytic function with non-negative Taylor coefficients ${c_n\ge 0}$ and A and $B^*$ are bounded quasinormal operators, such that $\max \{\vert {\!\,\!\vert {\,\!A\!\,}\vert \!\,\!}\vert ^2,\vert {\!\,\!\vert {\,\!B\!\,}\vert \!\,\!}\vert ^2\}$ is less than the radius of convergence for f,  then for all ${X\in {{\mathcal {C}}_{\Phi }({\mathcal H})}}$

$$\begin{aligned} \bigl \vert {\!\bigl \vert {\,\!f(A\;\!\!\otimes \!\,\!B)X\,\!}\bigr \vert \!}\bigr \vert _\Phi =\biggl \vert {\!\biggl \vert {\,\!\sum _{n=0}^\infty c_n A^n\!\,\!XB^n\,\!}\biggr \vert \!}\biggr \vert _\Phi \;\!\!\!\;\!\!\leqslant \bigl \vert {\!\bigl \vert {\,\!\!\sqrt{f(A^*\!\,\!A)}X \!\sqrt{f(B B^*)}\,\!}\bigr \vert \!}\bigr \vert _\Phi .\end{aligned}$$

If, in addition, $A,B^*$ are contractions and $0<\sum _{n=0}^{\infty }c_n<{+\;\!\!\infty },$ then

$$\begin{aligned}&\bigl \vert {\!\bigl \vert {\,\!\!\sqrt{I-A^*\!A}\bigl ({f(A)X-Xf(B)}\bigr )\,\!\!\sqrt{I-BB^*}\,\!}\bigr \vert \!}\bigr \vert _\Phi \\&\quad \leqslant \bigl \vert {\!\bigl \vert {\,\!\!\sqrt{f(1)I-f(A^*\!A)}{({AX-XB})}\,\!\!\sqrt{f(1)I-f(BB^*)}\,\!}\bigr \vert \!}\bigr \vert _\Phi \\&\quad \leqslant \bigl \vert {\!\bigl \vert {\,\!\!\sqrt{f(1)I-|f(A)|^2/f(1)}{({AX-XB})}\,\!\!\sqrt{f(1)I-|f(B^*)|^2/f(1)}\,\!}\bigr \vert \!}\bigr \vert _\Phi . \end{aligned}$$