arXiv subjects
Publications and source records attributed to Marzieh Mehrazin.
In this paper, we define a new concept of numerical range $W_{o}(\cdot)$ and prove its basic results. We also define the numerical radius $ω_{o}(\cdot)$ and prove that $$ω_{o}(T)\leq||| T|||\leq 2ω_{o}(T).$$
In this paper, we introduce a new type of parallelism for bounded linear operators on a Hilbert space $\big(\mathscr{H}, \langle \cdot ,\cdot \rangle\big)$ based on numerical radius. More precisely, we consider operators $T$ and $S$ which satisfy $ω(T + λS) = ω(T)+ω(S)$ for some complex unit $λ$. We show that $T \parallel_ω S$ if and only if there exists a sequence of unit vectors $\{x_n\}$ in $\mathscr{H}$ such that \begin{align*} \lim_{n\rightarrow\infty} \big|\langle Tx_n, x_n\rangle\langle Sx_n, x_n\rangle\big| = ω(T)ω(S). \end{align*} We then apply it to give some applications.