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

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

Asymmetric Fuglede Putnam's Theorem for operators reduced by their eigenspaces

Fuglede-Putnam Theorem have been proved for a considerably large number of class of operators. In this paper by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators.

Fuglede-Putnam type commutativity theorems for $ EP $ operators

Fuglede-Putnam theorem is not true in general for $ EP $ operators on Hilbert spaces. We prove that under some conditions the theorem holds good. If the adjoint operation is replaced by Moore-Penrose inverse in the theorem, we get Fuglede-Putnam type theorem for $ EP $ operators -- however proofs are totally different. Finally, interesting results on $ EP $ operators have been proved using several versions of Fuglede-Putnam type theorems for $ EP $ operators on Hilbert spaces.

Unbounded generalizations of the Fuglede-Putnam theorem and applications to the commutativity of self-adjoint operators

In this article, we prove and disprove several generalizations of unbounded versions of the Fuglede-Putnam theorem. As applications, we give conditions guaranteeing the commutativity of a bounded self-adjoint operator with an unbounded closed symmetric operator.

Unbounded generalizations of the Fuglede-Putnam theorem

In this paper, we prove and disprove several generalizations of unbounded versions of the Fuglede-Putnam theorem.

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$\|…