乘法算子的范数：回答Fialkow-Loebl问题
Norms of multiplication operators: answering Fialkow--Loebl question
浏览论文内容
中文总结 AI 辅助
该研究针对因子上的乘法算子，证明其值域包含于对称赋范算子空间的充要条件，给出其范数公式并回答了Fialkow-Loebl问题，还分析了弱$L_p$空间自然拟范数的非单调性。
中文摘要 AI 辅助
设$\uD835\udd00$是带有半有限忠实正规迹$\uD835\udd02$的因子，$E(0,\u221E)$是对称赋范函数空间，$E(\uD835\udd00,\uD835\udd02)$是对应的对称赋范算子空间。设$a,b$是与$\uD835\udd00$关联的$\uD835\udd02$-可测算子，证明乘法算子$S_{a,b}:x\to axb$在$\uD835\udd00$上的值域包含于$E(\uD835\udd00,\uD835\udd02)$当且仅当$\u03BC(a)\u03BC(b)$属于$E(0,\u221E)$，其中$\u03BC(x)$是与$\uD835\udd00$关联的$\uD835\udd02$-可测算子$x$的广义奇异值函数。此外，有$\u2016S_{a,b}\u2016_{\uD835\udd00\to E(\uD835\udd00,\uD835\udd02)} = \u2016\u03BC(a)\u03BC(b)\u2016_{E(0,\u221E)}$，这回答了Fialkow和Loebl在1984年提出的问题。还考虑了拟赋范情形，证明弱$L_p$空间（$0<p<\u221E$）的自然拟范数关于对数次优控制不是单调的。
英文摘要
Let $\mathcal{M}$ be a factor equipped with a semi-finite faithful normal trace $τ$. Let $E(0,\infty)$ be a symmetrically normed function space and $E(\mathcal{M},τ)$ be the corresponding symmetrically normed operator space. Suppose that $a, b$ are $τ$-measurable operators affiliated with $\mathcal{M}$. It is shown that the range of the multiplication operator $S_{a,b}: x\mapsto axb$ on $\mathcal{M}$ is contained in $E(\mathcal{M}, τ)$ if and only if $μ(a)μ(b)$ belongs to $E(0, \infty)$, where $μ(x)$ stands for the generalized singular value function of a $τ$-measurable operators $x$ affiliated with $\mathcal{M}$. Moreover, we have $$ \|S_{a,b}\|_{\mathcal{M}\to E(\mathcal{M},τ)}= \| μ(a )μ( b) \|_{E (0,\infty) }, $$ which answers a question by Fialkow and Loebl (1984). We also consider the quasi-normed case, and show that the natural quasi-norm of weak $L_p$-space, $0<p<\infty$, is not monotone with respect to the logarithmic submajorisation.
