局部可测算子的Fuglede-Putnam定理

Yet another generalization of the Fuglede-Putnam theorem to unbounded operators

In this note, we give the most natural (perhaps the simplest ever) generalization of the Fuglede-Putnam theorem where all operators involved are unbounded.

The Berberian's transform and an asymmetric Putnam-Fuglede theorem

We present how to apply a Berberian's technique to asymmetric Putnam-Fuglede theorems. In particular, we proved that if $A, B \in B(H)$ belong to the union of classes of $*$-paranormal operators, p-hyponormal operators, dominant operators and operators of class Y and $AX = XB^*$ for some $X \in B(H)$, then $A^*X = XB$. Moreover, we gave a new counterexample for an asymmetric Putnam-Fuglede theorem for paranormal operators

基于变换的Fuglede-Putnam型定理

Fuglede-Putnam type theorems via the Aluthge transform

令$A=U|A|$和$B=V|B|$是$A\in \mathbb{B}(\mathscr{H}_1)$和$B\in \mathbb{B}(\mathscr{H}_2)$的极分解，并令$Com(A,B)$代表算子集合$X\in\mathbb{B}(\mathscr{H}_2,\mathscr{H}_1)$，使得$AX=XB$。如果$Com(A,B)\subseteqCom(A^\ast,B^\ast)$，则称一对$(A,B)$具有FP性质。设$\tilde{C}$表示有界线性算子$C$的高斯变换.证明了：（i）若$A$和$B$可逆且$(A,B)$具有FP-性质，则$(\tilde{A},\tilde{B})$也可逆;（ii）若$A$和$B$可逆且$U$和$V$的谱都包含在某个开循环中且$(\tilde{A},\tilde{B})$具有FP-性质，则$(A,B)$也可逆;（iii）如果$(A,B)$具有FP性质，则为$Com(A,B)\subseteqCom(\tilde{A},\tilde{B})$，此外，如果$A$可逆，则为$Com(A,B)=Com(\tilde{A},\tilde{B})$。最后，如果$Re(U|A|^{1\over2})\geq a>0$和$Re(V|B|^{1\over2})\geq a>0$和$X$是一个算子，使得$U^* X=XV$，那么我们证明了$\|…

关于von Neumann algebras、Mahler测度和Ljapetry指数的行列式

Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents

对于概率空间上的遍历测度保持作用，考虑相应的交叉积vonNeumann代数。我们计算了Fuglede-Kadison行列式的一类运营商在此冯诺依曼代数的Ljapanese指数相关的可测上循环。这个证明是基于Dykema和Schultz最近的工作。作为一个应用，得到了离散海森堡群的冯·诺依曼代数中非交换多项式的Fuglede-Kadison行列式的公式。这些是林德和施密特通过熵的考虑得到的。