Further refinements of generalized numerical radius inequalities for Hilbert space operators

In this paper, we show some refinements of generalized numerical radius inequalities involving the Young and Heinz inequality. In particular, we present w_{p}^{p}(A_{1}^{*}T_{1}B_{1},\dots,A_{n}^{*}T_{n}B_{n})\leq\frac{n^{1-\frac{1% }{r}}}{2^{\frac{1}{r}}}\bigg{\|}\sum_{i=1}^{n}[B_{i}^{*}f^{2}(|T_{i}|)B_{i}]^{% rp}+[A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i}]^{rp}\bigg{\|}^{\frac{1}{r}}-\inf_{\|x\|% =1}\eta(x), where {T_{i},A_{i},B_{i}\in\mathbb{B}(\mathscr{H})} {(1\leq i\leq n)} , f and g are nonnegative continuous functions on {[0,\infty)} satisfying {f(t)g(t)=t} for all {t\in[0,\infty)} , {p,r\geq 1} , {N\in\mathbb{N}} , and \displaystyle\eta(x)=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{N}\Bigl{(}\sqrt[2^{j% }]{\big{\langle}(A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i})^{p}x,x\big{\rangle}^{2^{j-1% }-k_{j}}\big{\langle}(B_{i}^{*}f^{2}(|T_{i}|)B_{i})^{p}x,x\big{\rangle}^{k_{j}}} \displaystyle -\sqrt[2^{j}]{\big{\langle}(B_{i}^{*}f^{2}(|T_{i}|)B_{i}% )^{p}x,x\big{\rangle}^{k_{j}+1}\big{\langle}(A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i})% ^{p}x,x\big{\rangle}^{2^{j-1}-k_{j}-1}}\,\Big{)}^{2}.

A certain generalization of the Heinz inequality for positive operators

Norm inequalities of the form [Formula: see text] with [Formula: see text] and [Formula: see text] are studied. Here, [Formula: see text] are operators with [Formula: see text] and [Formula: see text] is an arbitrary unitarily invariant norm. We show that the inequality holds true if and only if [Formula: see text].

A Converse of the Loewner–Heinz Inequality, Geometric Mean and Spectral Order

Let A, B be non-negative bounded self-adjoint operators, and let a be a real number such that 0 < a < 1. The Loewner–Heinz inequality means that A ≤ B implies that Aa ≦ Ba. We show that A ≤ B if and only if (A + λ)a ≦ (B + λ)a for every λ > 0. We then apply this to the geometric mean and spectral order.

Some Properties of Furuta Type Inequalities and Applications

This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta type inequalities underA≥B≥0are obtained via Loewner-Heinz inequality; as an application, a proof of Furuta inequality is given without using the invertibility of operators. Secondly, we show a unified satellite theorem of grand Furuta inequality which is an extension of the results by Fujii et al. At the end, a kind of Riccati type operator equation is discussed via Furuta type inequalities.