Spectral Properties of Class of Sequences with Respect to Matrix Operators

Let us consider X to be a linear space, the set of all bounded linear operators defined on X into itself. We denote it by B(X). Let \(T\in B(X)\) and X is a Banach space then the adjoint operator \(T^*\) of T is defined as a bounded linear operator on the dual of X which is denoted by \(X^*\) and defined by \((T^*\varPhi )(x) = \varPhi (Tx)\) for all \(\varPhi \in X^*\) and \(x\in X\) . Let \(T\in B(X)\) and \(T :D(T) \rightarrow X\) generate a complex number \(\alpha \) of the operator \((T-\alpha I)\) defined on the domain D(T), which is denoted by \(T_{\alpha }\) . Then \((T-\alpha I)^{-1} = T_{\alpha }^{-1}\) is called the resolvent operator of \(T_{\alpha }\) . By default \(\alpha \) is a complex number such that The resolvent set denoted by \(\rho (T )\) is the set of all such values \(\alpha \) of T and its complement, written as \(\mathbb {C} \setminus \rho (T )\) is called the spectrum of the operator T denoted by \(\sigma (T ).\) The spectrum of an operator is partitioned into the following three disjoint sets: