Related papers: Determining elements in Banach algebras through sp…

Let $A$ be a complex unital Banach algebra with unit $1$. An element $a\in A$ is said to be of \textit{$G_{1}$-class} if $$\|(z-a)^{-1}\|=\frac{1}{\text{d}(z,\sigma(a))} \quad \forall z\in \mathbb{C}\setminus \sigma(a).$$ Here $d(z,…

We study algebraic properties of full rank 1 algebras in a general framework and derive a method to verify if one such matrix polynomial sub-algebra is bispectral. We give two examples illustrating the method. In the first one, we consider…

Let $X$ be a locally compact Hausdorff space, and $A$ be a commutative semisimple Banach algebra over the scalar field $\mathbb{C}$. The correlation between different types of BSE- Banach algebras $A$, and the Banach algebra $C_{0}(X, A)$…

Adapting the notion of the spectrum $\Sigma_a$ for an element $a$ in an ultrametric Banach algebra (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum $\sigma^{Ber}_R(u)$ of an element $u$ in a Banach ring $R$.…

Sylvester equations $AX-XB=C$ have unique solutions for all $C$ when the spectra of $A$ and $B$ are disjoint. Here $A$ and $B$ are bounded operators in Banach spaces. We discuss the existence of polynomials $p$ such that the spectra of…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded…

We give an example of a positive element $a$ in some ordered Banach algebra $A$ such that its spectrum is equal to $\{1\}$ and it is not greater than or equal to the unit element of $A$.

Given a Banach space $A$ and fix a non-zero $\varphi\in A^*$ with $\|\varphi\|\leq 1$. Then the product $a\cdot b=\langle\varphi, a\rangle\ b$ turning $A$ into a Banach algebra which will be denoted by $_\varphi A.$ Some of the main…

We extend an example of B. Aupetit, which illustrates spectral discontinuity for operators on an infinite dimensional separable Hilbert space, to a general spectral discontinuity result in abstract Banach algebras. This can then be used to…

For given unital complex Banach algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, let $\mathfrak{M}$ be a Banach module acting between them. Let $a\in \mathcal{A}_1$, $b\in\mathcal{A}_2$, and $c\in\mathfrak{M}$ be provided such that…

Suppose that ${\mathcal A}$ is an algebra, $\sigma,\tau:{\mathcal A}\to{\mathcal A}$ are two linear mappings such that both $\sigma({\mathcal A})$ and $\tau({\mathcal A})$ are subalgebras of ${\mathcal A}$ and ${\mathcal X}$ is a…

Let $A$ be a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map into $A$, we derive a generalization of Gelfand's famous Power Boundedness Theorem.…

In this paper we characterize the Birkhoff--James orthogonality with respect to the numerical radius norm $v(\cdot)$ in $C^*$-algebras. More precisely, for two elements $a, b$ in a $C^*$-algebra $\mathfrak{A}$, we show that $a\perp_{B}^{v}…

For a positive element $A$ of a $C^*$-algebra $\mathfrak{A}$, let ${\|X\|}_{A}$ denote the $A$-operator semi-norm of $X\in\mathfrak{A}$. In this paper, we aim to introduce and study the notion of $A$-spectrum for $X$ such that…

Let $A$ be a non-unital Banach algebra and let $A_e = A \oplus {\mathbb C}1$ be the unitization of $A$. It is true that if $A_e$ has the spectral extension property (SEP), then $A$ has the same. Does the converse hold? In this paper, we…

Let $A$ be a unital Banach algebra over $\mathbb C$, and suppose that the nonzero spectral values of, respectively, $a\mbox{ and }b\in A$ are discrete sets which cluster at $0\in\mathbb C$, if anywhere. We develop a plane geometric formula…

We initiate the study of cortex algebras (commutative Banach algebras with extendable multiplicative-linear functionals) and the spectral permanence property. In addition, we analyze some particular examples in this context and present open…

Let $A$ be a commutative semisimple Arens regular unital Banach algebra. The correlation between the BSE-property of the Banach algebra $A$ and its second duals are assessed. It is found and approved that if $A$ is a BSE-algebra, then so is…

Let $\mathbb B_n$ be the open unit ball in $\mathbb C^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on $\mathbb B_n$ satisfying some general conditions. These spaces include…

We study bounded bilinear maps on a C$^*$-algebra $A$ having product property at $c\in A$. This leads us to the question of when a C$^*$-algebra is determined by products at $c.$ In the first part of our paper, we investigate this question…