论文标题
C* - 模块运算符，在广义的cauchy--schwarz类型不平等中满足
C*-module operators which satisfy in the generalized Cauchy--Schwarz type inequality
论文作者
论文摘要
令$ \ Mathcal {l}（\ Mathscr {h}）$表示$ C^*$ - 在Hilbert $ c^*$ - Module $ \ Mathscr {h h} $上的可伴随运算符的代数。我们在$ \ Mathcal {l}（\ Mathscr {h}）$中介绍了对运营商的广义cauchy-schwarz $，并研究了操作员的各种属性，这些属性满足了概述的cauchy-schwarz不平等。特别是，我们证明，如果运算符$ a \ in \ mathcal {l}（\ mathscr {h}）$满足了概括的cauchy-schwarz不平等，使$ a $ a $具有极地分解，那么$ a $ a $是超凡的。此外，我们表明，如果对于$ a $，平等在广义的Cauchy-Schwarz不平等中都有，那么$ a $是同伴。除其他事项外，当$ a $具有极地分解时，我们证明$ a $是半高调的，并且仅当$ \ big \ | \ | \ langle ax，y \ rangle \ big \ | \ leq \ big \ | {| a |}^{1/2} x \ big \ | | \ big \ | {| a | a |}^{1/2} y \ big \ | $ for ALL $ x，y \ in \ MATHSCR {H} $。
Let $\mathcal{L}(\mathscr{H})$ denote the $C^*$-algebra of adjointable operators on a Hilbert $C^*$-module $\mathscr{H}$. We introduce the generalized Cauchy-Schwarz inequality for operators in $\mathcal{L}(\mathscr{H})$ and investigate various properties of operators which satisfy the generalized Cauchy--Schwarz inequality. In particular, we prove that if an operator $A\in\mathcal{L}(\mathscr{H})$ satisfies the generalized Cauchy-Schwarz inequality such that $A$ has the polar decomposition, then $A$ is paranormal. In addition, we show that if for $A$ the equality holds in the generalized Cauchy-Schwarz inequality, then $A$ is cohyponormal. Among other things, when $A$ has the polar decomposition, we prove that $A$ is semi-hyponormal if and only if $\big\|\langle Ax, y\rangle\big\| \leq \big\|{|A|}^{1/2}x\big\|\big\|{|A|}^{1/2}y\big\|$ for all $x, y \in\mathscr{H}$.
