For a separable Banach space X, let L denotes the algebra of bounded Operators on X endowed with the strong Operator topology. In this note we provide a sufficient condition, in terms of open set in the strong Operator topology, for a bounded linear mapping acting on L to be supercyclic with this topology. The main result has some useful consequences and make easy the proof of some related results.

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In this paper, based on the results and theorems of C^*-algebra valued S_b-metric spaces, we solve one type of matrix Operator equations in L(H) defined by X-∑ _(n=1)^∞ ▒ 〖 A_n^* XA_n=Q〗 , in which H is a Hilbert space and L(H) is the set of linear and bounded Operators on H. Also, we prove that the Kannan contraction mapping has a unique fixed point in C^*-algebra valued S_b-metric spaces. Furthermore, we prove that the Chatterjea-type contraction mapping has a unique fixed point in C^*-algebra valued S_b-metric spaces. Finally, by using the Banach contraction principle in C^*-algebra valued S_b-metric spaces that has previously been studied by the authors of this article and by using the results of the above theorems, we solve the above matrix Operator equation in C^*-algebra valued S_b-metric spaces. As well as, we show that this matrix Operator equation has a unique solution in L(H), and this solution is a Hermitian Operator.

The main goal of this paper is to introduce analogously modal Operators on $L$-algebras and study their properties. To begin with, we introduce the notion of modal Operators on $L$-algebras and investigate some important properties of this Operator. In order for the kernel of modal Operator to be ideal, we investigate what conditions are required. Relations between modal Operator and endomorphism of $L$-algebras are investigated. Also, we define the concept of positive $L$-algebra and some characterizations of positive $L$-algebra are established. Finally, we introduce a map $k_{a}$ and show that $k_{a}$ is a modal Operator and we prove that the set of all $k_{a}$ on a positive $L$-algebra makes a dual BCK-algebra.

It is known that even duals of a Banach algebra A with one of Arens products are Banach algebras, these products are natu-ral multiplications extending the one on A. But the essence of A ∗,is completely di, erent. By de, ning new products, we investigate some algebraic and spectral properties of odd duals of A. We will show re-lations between these products and Arens products, weak or weak-star continuity, commutativity and unit elements of these algebras. We also determine the spectrum and multiplier algebra for A ∗, , and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of algebras.