In this article, we study special $ 2 imes n $ 2 & times;n row Suslin matrices $ S_n<^>2(V,W) $ Sn2(V,W), arising from the pair of matrices $ V,W \in M_{2 ,n}(R) $ V,W is an element of M2,n(R) with $ V.W<^>T = I_2 $ V.WT=I2. These matrices generate a subgroup $ SUm_n<^>2(R) $ SUmn2(R) of the general linear group $ GL_{2<^>{2n-1}}(R) $ GL22n-1(R). Our primary focus is on the elementary subgroup $ EUm_n<^>2(R) \subset SL_{2<^>{2n-1}}(R) $ EUmn2(R)subset of SL22n-1(R) of $ SUm_n<^>2(R) $ SUmn2(R), for which we determine a complete set of generators by establishing the key lemma. We further investigate the algebraic properties of these generators and prove a structure theorem for $ EUm_n<^>2(R) $ EUmn2(R) that simplifies the generating set. The findings presented in this article enhance the understanding of matrix group generation and have potential implications for algebraic K-theory and related areas.
Assume that M is a Hilbert C*-module over a C* -algebra A . In this paper, we first propose the concepts of generalized derivations and generalized Jordan derivations on M , and then by giving some useful properties of M , we prove that C -linear generalized (Jordan) derivations on M are automatically continuous under the norm topology. In addition, we show that, for any C -linear map delta on M , if A is commutative, then the following statements are equivalent: (1) delta is a generalized Jordan derivation, (2) delta is a generalized derivation, (3) delta is a Jordan derivation, (4) delta is a derivation; if delta is A -linear, M is full and A is unital, then delta is a generalized derivation if and only if delta is a derivation. Finally, we also discuss the structure of other generalized forms of (Jordan) derivations.
We construct the sequence spaces h(Delta((2))(q)) = h(Delta q(2)) and bv(Delta((2))(q)) = bv ((2))(Delta q), where Delta((2))(q) is a second order q-difference operator defined by (Delta((2))(q)u)(d) = u(d) - (1 + q)u(d- 1) + qu(d-2) for all d is an element of Z(0)(+), where we presume that u(d) = 0 for d <= 0. We obtain some inclusion relations, Schauder bases, and important duals of the spaces h(Delta((2))(q)) and bv(Delta((2))(q)). We also determine necessary and sufficient condition for a matrix to map between the spaces h(Delta((2))(q)) /bv(Delta((2))(q)) and any arbitrary space mu. In the final section, we determine the point spectrum and the spectrum of the operator Delta((2))(q) over the space h.
In this paper we establish necessary and sufficient condition for a signed Cayley graph to be strongly regular and present several constructions of strongly regular signed Cayley graphs, which are proven to be net-regular. Furthermore, we calculate their adjacency eigenvalues and demonstrate that they have at most four distinct eigenvalues.
This article examines a way to define the notion of left (right) $ (b,c) $ (b,c)-polar elements in an associative ring R. Necessary and sufficient conditions of an element $ a\in \mathcal {R} $ a is an element of R to be left (right) $ (b,c) $ (b,c)-polar are investigated. We show that an element $ a\in \mathcal {R} $ a is an element of R is left (right) $ (b,c) $ (b,c)-polar if and only if a is strongly left (right) $ (b,c) $ (b,c)-invertible. Moreover, the one-sided $ (b,c) $ (b,c)-polarity is characterized in terms of left and right annihilators. As an application, we provide a characterization of the one sided core inverse. Further results and properties are obtained.
