The Spatial Numerical Range in Non-unital, Normed Algebras and Their Unitizations
Year: 2025
Type: article
Abstract: Let $(A,\|\cdot\|)$ be any normed algebra (not necessarily complete nor unital). Let $a \in A$ and let $V_A(a)$ denote the spatial numerical range of $a$ in $(A,\|\cdot\|)$. Let $A_e = A + \mathbb{C}1$ be the unitization of $A$. If $A$ is faithful, then we get two norms on $A_e$; namely, the operator norm $\|\cdot\|_{\text{op}}$ and the $\ell_1$-norm $\|\cdot\|_1$. Let $A_{\text{op}} = (A,\|\cdot\|_{\text{op}})$, $A_{\text{op},e} = (A_e,\|\cdot\|_{\text{op}})$, and $A_{1,e} = (A_e,\|\cdot\|_1)$. We can calculate the spatial numerical range of $a$ in all three normed algebras. Because the spatial numerical range highly depends on the identity as well as on the completeness and the regularity of the norm, they are different. In this paper, we study the relations among them. Some results that are proved in \cite{ref2}, Section 2, and \cite{ref3}, Section 10, will become corollaries of our results. We shall also show that the completeness and regularity of the norm is not required in \cite{ref6}, Theorem 2.3.
Source: SARAJEVO JOURNAL OF MATHEMATICS
DOI: 10.5644/sjm.20.02.08
Authors: Haresh V. Dedania, Avadh B. Patel
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4409439110