Extensions of the operator Bellman and operator Holder type inequalities

Authors : Mojtaba Bakherad, Fuad Kıttaneh

Pages : 12-29

Doi:10.33205/cma.1435944

View : 99 | Download : 216

Publication Date : 2024-03-15

Article Type : Research Paper

Abstract :In this paper, we employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator H\\\"{o}lder type inequalities. Among other results, it is shown that if $\\mathbf{A}=(A_t)_{t\\in \\Omega}$ and $\\mathbf{B}=(B_t)_{t\\in \\Omega}$ are continuous fields of positive invertible operators in a unital $C^*$-algebra ${\\mathscr A}$ such that $\\int_{\\Omega}A_t\\,d\\mu(t)\\leq I_{\\mathscr A}$ and $\\int_{\\Omega}B_t\\,d\\mu(t)\\leq I_{\\mathscr A}$, and if $\\omega_f$ is an arbitrary operator mean with the representing function $f$, then \\begin{align*} \\left(I_{\\mathscr A}-\\int_{\\Omega}(A_t \\omega_f B_t)\\,d\\mu(t)\\right)^p \\geq\\left(I_{\\mathscr A}-\\int_{\\Omega}A_t\\,d\\mu(t)\\right) \\omega_{f^p}\\left(I_{\\mathscr A}-\\int_{\\Omega}B_t\\,d\\mu(t)\\right) \\end{align*} for all $0 < p \\leq 1$, which is an extension of the operator Bellman inequality.
Keywords : Bellman inequality, Cauchy Schwarz inequality, H\\\ o lder inequality, operator mean, Hadamard product, continuous field of operators, C^ algebra