2025-12-29
Mohammad Javad Mehdipour, Narjes Salkhordeh
In this paper, we investigate $(p, q)$-derivations in both general algebras and Banach algebras. Our main result extends the Singer-Wermer theorem to $(p, q)$-derivations, proving that their ranges are contained in the radical of the algebra. As a direct consequence, we prove that the range of any left derivation lies within the radical of the algebra even without assuming continuity. This improves a result due to Bra\v{s}er and Vukman. For Banach algebras, we show that primitive ideals remain invariant under bounded $(p, q)$-derivations. Finally, we study $(p, q)$-derivations of group algebras and give an answer to the question posed.
2025-12-23
Nasrin Ebrahimzadeh Eghbal, Elnaz Osgooei, Mohammad Hossein Sattari
This paper presents a new definition for g-frames in Kreinspaces, based on the properties of Krein Space. It is shown that a g-frame in a Krein space is also a g-frame in the associated Hilbertspace. However, the converse is not true unless the span of the J-adjoint of positive operators is uniformly J-positive, and the span ofthe J-adjoint of negative operators is uniformly J-negative. The J-g-frame operator is defined for g-frames in Krein spaces, and its properties,such as invertibility and boundedness, are discussed. Additionally, a J-g-dual sequence is introduced for g-frames in Krein spaces and shownto be a g-frame in the associated Hilbert space, but not necessarily inthe Krein space. Finally, the relationship between a g-frame in a Kreinspace and the sequence induced by it in the associated Hilbert space isexamined.
2025-09-10
Omid Rahmanian, Prof. Ali Akbar Estaji, Mostafa Abedi
We introduce pointfree counterparts of countably disconnectivity, specifically countably basically disconnected frames (briefly, $c$-basically disconnected frames) and countably $F$-frames (briefly, $F_c$-frames), along with their frame-theoretic characterizations. We present characterizations of zero-dimensional extremally disconnected frames, zero-dimensional $c$-basically disconnected frames, and zero-dimensional $F_c$-frames $L$ using the ring-theoretic properties of the ring $\mathcal{R}_cL$ of continuous real-valued functions with countable images on $L$. We show that a zero-dimensional frame $L$ is extremally disconnected if and only if any annihilator ideal of $\mathcal{R}_cL$ is generated by an idempotent, $c$-basically disconnected if and only if the annihilator of each element of $\mathcal{R}_cL$ is generated by an idempotent, and an $F_c$-frame if and only if the annihilator of each element of $\mathcal{R}_cL$ is pure.
2025-06-02
Fethiye Müge Sakar, Wasim Ul Haq, Faseeh U Rahman
In this present paper,we consider a subclass of close-to-convex functions associated with a vertical strip domain. We obtain several results concerned with integral coefficient inequalities,convolutions, and representations for functions belonging to this class $ \mathbf{MC_c}$. Furthermore, we consider the Fekete-Szego, inclusion relations and radius problems involving certain classes of strongly close-to-convex functions, parabolic close-to-convex functions, and other types of close-to-convex functions.