Let $(\mathcal{M},\tau )$ be a semi-finite von Neumann algebra, $L_0\left(\mathcal{M}\right)$ be the set of all τ-measurable operators, ${\mu }_t\left(x\right)$ be the generalized singular number of $x\in L_0\left(\mathcal{M}\right)$ and $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a concave function. We proved that if $x_1,x_2,\dots ,x_n$ are normal operators in $L_0\left(\mathcal{M}\right)$, then $\mu \left(f\right(|\sum _{k=1}^n{x}_k|\left)\right)$ is submajorized by $\mu \left(f\right(\sum _{k=1}^n|x_k|\left)\right)$. As an application, we obtained that if x is a matrix of normal operators $x_{ij}$ in $L_0\left(\mathcal{M}\right)$, then $\mu \left(f\right(|x|\left)\right)$ is submajorized by $\mu \left(\sum _{i,j=1}^{n}f\right(|x_{ij}|\left)\right)$.

中文翻译：

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τ-可测算子矩阵的次主要不等式

让$(\mathcal{米},\tau )$是一个半有限冯诺依曼代数，${\mathrm{大号}}_0\left(\mathcal{米}\right)$是所有τ可测算子的集合，${\mu }_{吨}\left(X\right)$是的广义奇异数$X\in {\mathrm{大号}}_0\left(\mathcal{米}\right)$和$F：[0,\mathrm{\infty })\to [0,\mathrm{\infty })$为凹函数。我们证明了如果$X_1,X_2,\dots ,X_n$是正常的运算符${\mathrm{大号}}_0\left(\mathcal{米}\right)$， 然后$\mu \left(F\right(|\sum _{ķ=1}^n{X}_{ķ}|\left)\right)$次要于$\mu \left(F\right(\sum _{ķ=1}^n|X_{ķ}|\left)\right)$. 作为一个应用程序，我们得到如果x是一个正规算子的矩阵$X_{\mathrm{一世}j}$在${\mathrm{大号}}_0\left(\mathcal{米}\right)$， 然后$\mu \left(F\right(|X|\left)\right)$次要于$\mu \left(\sum _{\mathrm{一世},j=1}^{n}F\right(|X_{\mathrm{一世}j}|\left)\right)$.