Vol. 30 (2024), no. 3-4 (Current Issue)

A bounded operator $T$ in a Banach space $X$ is said to satisfy the essential descent spectrum equality, if the descent spectrum of $T$ coincides with the essential descent spectrum of $T$. In this note, we give some conditions under which the equality $\sigma_{desc}(T) = \sigma^e_{desc}(T)$ holds for $T$.

This paper delves into the investigation of compact sets within the complex plane under a boundary constraint. Specifically, we focus on scenarios where a compact set $A$ is enclosed by a curve $\partial A$ that lies within the boundary of the augmented unit disk $\partial \mathbb{D}$, including the origin. The main goal is to establish a theorem that characterizes the possible configurations of such sets. By interweaving the principles of topology and operator theory, this study not only enhances our comprehension of compact sets under specialized boundary conditions but also underscores a practical implication in the realm of operator theory. This connection is particularly evident in the examination of the spectrum of operators that meet specific isometric conditions.

In this work, we prove the generalised Hyer Ulam stability of the following functional equation \begin{equation} \phi(x)+\phi(y)+\phi(z)=q \phi\left(\sqrt[s]{\frac{x^s+y^s+z^s}{q}}\right),\qquad |q| \leq 1 \end{equation} and $s$ is an odd integer such that $s\geq 3$, in modular space, using the direct method, and the fixed point theorem.

We compute the resolvent operator of the generalized Sub-Laplacian.

In this paper, we prove the generalization of Titchmarsh's theorem for the second Hankel-Clifford transform for functions satisfying the $(m, \mu, 2)$-Hankel-Clifford Lipschitz condition in the space $\mathrm{L}^{2}((0,+\infty), x^{\mu})$, where $\mu\geq 0$.

This paper employs $C^0-$arguments to study the action of the identity component, topologized with the $C^\infty$ Whitney topology, $Diff^{\Omega,\infty}_{id}(M) $ in the group of volume-pre\-serving diffeomorphisms on the space $\mathcal Z^1(M)$ of all closed $1-$forms on a compact connected oriented manifold $(M, \Omega)$. When $M$ is closed, we recover that $Diff^{\Omega,\infty}_{id}(M) $ is $C^0-$closed in the group $Diff^{\infty}(M) $ of all smooth diffeomorphisms of $M$. This implies that in two dimensions, the identity component in the group of symplectomorphisms is $C^0-$closed. We discuss several applications in the context of $C^0$ symplectic geometry for Lefschetz closed symplectic manifolds. This includes an attempt to solve the $C^0$ flux conjecture.