Singular value and unitarily invariant norm inequalities for sums and products of operators

Abstract

In this note, we mainly investigate singular value and unitarily invariant norm inequalities for sums and products of operators. First, we present singular value inequality for the quantity $AX+YB$: let A, B, X and $Y \in B({\mathcal {H}})$ such that both A and B are positive operators. Then

$$\begin{aligned} s_{j}\left( (AX+YB)\oplus 0\right) \le s_{j}\left( (K+M)\oplus (L_{ 1} + N)\right) , \end{aligned}$$

for $j = 1,2,\ldots$, where $K=\frac{1}{2}A+\frac{1}{2}A^{\frac{1}{2}}|X^{*}|^{2}A^{\frac{1}{2}}$, $L_{1}=\frac{1}{2}B+\frac{1}{2}B^{\frac{1}{2}}|Y|^{2}B^{\frac{1}{2}}$, $M=\frac{1}{2}|B^{\frac{1}{2}}(X+Y)^{*}A^{\frac{1}{2}}|$ and $N=\frac{1}{2}|A^{\frac{1}{2}}(X+Y)B^{\frac{1}{2}}|$. In addition, based on the above singular value inequality, we establish a unitarily invariant norm inequality for concave functions. These results generalize inequalities obtained by Audeh directly. Finally, we present another more general singular value inequality for $\sum \nolimits _{i=1}^{m}A_{i}^{*}X_{i}^{*}B_{i}$:

$$\begin{aligned} s_{j}\left( \sum \limits _{i=1}^{m}A_{i}^{*}X_{i}^{*}B_{i}\oplus 0\right) \le s_{j} \left( \left( \sum \limits _{i=1}^{m}A_{i}^{*}f_{i}^{2}(|X_{i}|)A_{i}\right) \oplus \left( \sum \limits _{i=1}^{m}B_{i}^{*}g_{i}^{2}(|X_{i}^{*}|)B_{i}\right) \right) , \end{aligned}$$

for $j=1,2,\ldots$, where $A_{i}$, $B_{i}$ and $X_{i}\in B({\mathcal {H}})$ such that $A_{i}$ and $B_{i}$ ($i=1,2,\ldots ,m$) are compact operators and $f_{i}$, $g_{i}$ ($i=1,2,\ldots ,m$) are 2m nonnegative continuous functions on $[0,+\infty )$ with $f_{i}(t)g_{i}(t)=t$ ($i=1,2,\ldots ,m$) for $t\in [0,+\infty )$.