τ-可测正算子生成的交换子的若干不等式

Some Inequalities for Commutators of τ-Measurable Positive Operators

设M是具有正规忠实的半有限迹τ的von Neumann代数,‖.‖ρ是任意非交换Banach函数空间范数,‖.‖是M上的通常范数.证明了若A和B是τ-可测正算子,X∈M,则‖AX-XB‖ρ≤‖X‖‖AB‖ρ.还证明了若A,B是M中的正算子,X是τ-可测算子,则‖AX-XB‖ρ≤max(‖A‖,‖B‖)‖X‖ρ.由此得到了若A∈M是正算子,X是τ-可测正算子,则‖AX-XA‖ρ≤1/2‖A‖‖XX‖ρ.

Let M be a yon Neuman algebra equipped with a normal semifinite faithful trace τ. Let ｜｜·｜｜ρ is any non-commutative Banach function space norm and ｜｜·｜｜ is the usual operator norm in M We proved that if A and B are τ -measurable positive operators, and X X∈M then ｜｜AX-XB｜｜ρ≤｜｜X｜｜ ｜｜A＋B｜｜ρ. Also proved that if A and B are positive operators in M and X is τ -measurable operator, then ｜｜AX-XB｜｜ρ,≤max（｜｜A｜｜,｜｜B｜｜）｜｜X｜｜ρ. Consequently, if A ∈ M is positive operator and X is τ -measurable positive operator, then ｜｜AX-XA｜｜ρ≤1/2｜｜A｜｜ ｜｜X＋X｜｜ρ.