Some operator inequalities for unitarily invariant norms
Denote by \(\mathbf{B}(\mathcal{H})\) the algebra of all bounded linear operators on a complex separable Hilbert space \(\mathcal{H}\). For \(A\in \mathbf{B}(\mathcal{H})\), \(A\geq 0\) means that \(A\) is positive and \(A > 0\) means that \(A\) is invertible and positive. Let \(A, B, X \in M_{n}\) with \(A, B > 0\). The norm inequality \[ (2 + t){| \kern -0.25ex | \kern -0.25ex | A^{r}XB^{2 - r} + A^{2 - r}XB^{r} | \kern -0.25ex | \kern -0.25ex | } \leq 2 {| \kern -0.25ex | \kern -0.25ex | A^{2}X + tAXB + XB^{2} | \kern -0.25ex | \kern -0.25ex | } \tag{1} \] holds for every unitarily invariant norm \({| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }\), \(-2< t\leq 2\) and \(\frac{1}{2}\leq r\leq \frac{3}{2}\), see [\textit{X.-Z. Zhan}, SIAM J. Matrix Anal. Appl. 20, No. 2, 466--470 (1998; Zbl 0921.15011)]. Another important norm inequality is the well-known Cauchy-Schwarz type inequality: for every unitarily invariant norm \({| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }\) and every positive real number \(r\), the function \[ g(t) = {| \kern -0.25ex | \kern -0.25ex | |A^{t}XB^{1 - t}|^{r} | \kern -0.25ex | \kern -0.25ex | } \cdot {| \kern -0.25ex | \kern -0.25ex | |A^{1 - t}XB^{t}|^{r} | \kern -0.25ex | \kern -0.25ex | } \tag{2} \] is convex on the interval \([0, 1]\) and attains its minimum at \(t = \frac{1}{2}\). Assume now that \({| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }\) is a unitarily invariant norm on a two-sided ideal \(K_{{| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }}\) in \(\mathbf{B}(\mathcal{H})\). The following inequality is a consequence of the norm inequalities of the Cauchy-Schwarz type. Let \(p\) and \(q\) be two positive real numbers with \(p > 0\) or \(q > 0\). Then \[ {| \kern -0.25ex | \kern -0.25ex | |A^{p}XB^{q}|^{r} | \kern -0.25ex | \kern -0.25ex | } \cdot {| \kern -0.25ex | \kern -0.25ex | |A^{q}XB^{p}|^{r} | \kern -0.25ex | \kern -0.25ex | } \leq {| \kern -0.25ex | \kern -0.25ex | |A^{p + q}X|^{r} | \kern -0.25ex | \kern -0.25ex | } \cdot {| \kern -0.25ex | \kern -0.25ex | |XB^{p+q}|^{r} | \kern -0.25ex | \kern -0.25ex | }.\tag{3} \] In this paper, the authors present a generalization of the norm inequality (1) and some operator inequalities of Cauchy-Schwarz type: Theorem 1. Let \(A, B, X \in \mathbf{B}(\mathcal{H})\) with \(A, B > 0\) and \(X \in K_{{| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }}\) and \(p\geq q\geq r\geq 0\). Then the inequality \[ \begin{aligned} & (2 + t){| \kern -0.25ex | \kern -0.25ex | A^{\frac{3p + q}{2}}XB^{\frac{3q + p}{2}} + A^{\frac{3q + p}{2}}XB^{\frac{3p + q}{2}} | \kern -0.25ex | \kern -0.25ex | }\\ & \leq 4 {| \kern -0.25ex | \kern -0.25ex | A^{\frac{3p + q + 2r}{2}}XB^{\frac{3q + p - 2r}{2}} + A^{\frac{3q + p - 2r}{2}}XB^{\frac{3p + q + 2r}{2}} | \kern -0.25ex | \kern -0.25ex | } - 2(2 - t) {| \kern -0.25ex | \kern -0.25ex | A^{p+q}XB^{p+q} | \kern -0.25ex | \kern -0.25ex | }\\ & \leq {| \kern -0.25ex | \kern -0.25ex | A^{2(p+q)}X + XB^{2(p+q)} + tA^{p+q}XB^{p+q} | \kern -0.25ex | \kern -0.25ex | }\end{aligned} \] holds for any unitarily invariant norm \({| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }\) and \(t \in (-2, 2]\). As a consequence of Theorem~1, if \(A, B, X \in \mathbf{B}(\mathcal{H})\) with \(A, B > 0\), \(X \in K_{{| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }}\) and \(p, q> 0\), then the inequality \[ \begin{aligned} & (2 + t){| \kern -0.25ex | \kern -0.25ex | A^{\frac{3p + q}{2}}XB^{\frac{3q + p}{2}} + A^{\frac{3q + p}{2}}XB^{\frac{3p + q}{2}} | \kern -0.25ex | \kern -0.25ex | }\\ & \leq 2 {| \kern -0.25ex | \kern -0.25ex | A^{2(p+q)}X + XB^{2(p+q)} + tA^{p+q}XB^{p+q} | \kern -0.25ex | \kern -0.25ex | } \end{aligned} \] holds for any unitarily invariant norm \({| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }\) and \(t \in (-2, 2]\). Theorem 2. Let \(A, B, X \in \mathbf{B}(\mathcal{H})\) with \(A, B > 0\), \(X \in K_{{| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }}\) and \(p\geq q\geq s\geq 0\). Then the inequality \[ {| \kern -0.25ex | \kern -0.25ex | |A^{p}XB^{q}|^{r} | \kern -0.25ex | \kern -0.25ex | } \cdot {| \kern -0.25ex | \kern -0.25ex | |A^{q}XB^{p}|^{r} | \kern -0.25ex | \kern -0.25ex | } \leq {| \kern -0.25ex | \kern -0.25ex | |A^{p + s}XB^{q-s}|^{r} | \kern -0.25ex | \kern -0.25ex | } \cdot {| \kern -0.25ex | \kern -0.25ex | |A^{q-s}XB^{p+s}|^{r} | \kern -0.25ex | \kern -0.25ex | } \] holds for any unitarily invariant norm \({| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }\) and \(r>0\). Theorem 3. Let \(A, B, X \in \mathbf{B}(\mathcal{H})\) with \(A, B > 0\), \(X \in K_{{| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }}\), \(p\geq q\geq s\geq 0\) and \(r>0\). Then the function \[ f(s) = {| \kern -0.25ex | \kern -0.25ex | |A^{p + s}XB^{q-s}|^{r} | \kern -0.25ex | \kern -0.25ex | } \cdot {| \kern -0.25ex | \kern -0.25ex | |A^{q-s}XB^{p+s}|^{r} | \kern -0.25ex | \kern -0.25ex | } \] is increasing on \([0, q]\). As a consequence of Theorem 3, if \(A, B, X \in \mathbf{B}(\mathcal{H})\) with \(A, B > 0\), \(X \in K_{{| \kern -0.25ex | \kern -0.25ex | \cdot | \kern -0.25ex | \kern -0.25ex | }}\), \(t \in [0, 1]\) and \(r> 0\), then the function \[ g(t) = {| \kern -0.25ex | \kern -0.25ex | |A^{t}XB^{1-t}|^{r} | \kern -0.25ex | \kern -0.25ex | } \cdot {| \kern -0.25ex | \kern -0.25ex | |A^{1-t}XB^{t}|^{r} | \kern -0.25ex | \kern -0.25ex | } \] is decreasing on \([0, \frac{1}{2}]\) and increasing on \([\frac{1}{2}, 1]\).
scientific article; zbMATH DE number 6694412