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Numerical Radius Parallelism of Hilbert Space Operators
Bulletin of the Iranian Mathematical Society ( IF 0.7 ) Pub Date : 2019-10-20 , DOI: 10.1007/s41980-019-00295-3
Marzieh Mehrazin , Maryam Amyari , Ali Zamani

In this paper, we study the numerical radius parallelism for bounded linear operators on a Hilbert space \(\big ({\mathscr {H}}, \langle \cdot ,\cdot \rangle \big )\). More precisely, we consider bounded linear operators T and S which satisfy \(\omega (T + \lambda S) = \omega (T)+\omega (S)\) for some complex unit \(\lambda \), and is denoted by \(T \parallel _{\omega } S\). We show that \(T \parallel _{\omega } S\) if and only if there exists a sequence of unit vectors \(\{x_n\}\) in \({\mathscr {H}}\) such that$$\begin{aligned} \displaystyle {\lim _{n\rightarrow \infty }}\big |\langle Tx_n, x_n\rangle \langle Sx_n, x_n\rangle \big | = \omega (T)\omega (S). \end{aligned}$$We then apply it to give some applications.

中文翻译:

希尔伯特空间算子的数值半径并行性

在本文中,我们研究了希尔伯特空间\(\ big({\ mathscr {H}},\ langle \ cdot,\ cdot \ rangle \ big)\)上有界线性算子的数值半径并行性。更精确地,我们考虑有界的线性算子Ť小号满足\(\欧米加(T + \拉姆达S)= \欧米加(T)+ \欧米加(S)\)对于一些复杂的单元\(\拉姆达\),和用\(T \ parallel _ {\ omega} S \)表示。我们证明只有当\({\ mathscr {H}} \)中存在单位矢量序列\(\ {x_n \} \)时,\(T \ parallel _ {\ omega} S \)使得$$ \ begin {aligned} \ displaystyle {\ lim _ {n \ rightarrow \ infty}} \ big | \ langle Tx_n,x_n \ rangle \ langle Sx_n,x_n \ rangle \ big | = \ omega(T)\ omega(S)。\ end {aligned} $$然后我们将其应用于某些应用程序。
更新日期:2019-10-20
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