Abstract
The main purpose of this paper is to determine the fine spectrum with respect to Goldberg's classification of the operator defined by the lambda matrix over the sequence spaces andc. As a new development, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator on the sequence spaces andc. Finally, we present a Mercerian theorem. Since the matrix is reduced to a regular matrix depending on the choice of the sequence having certain properties and its spectrum is firstly investigated, our work is new and the results are comprehensive.
Highlights
Let X and Y be Banach spaces, and let T : X → Y be a bounded linear operator
By the statement “T is invertible,” it is meant that there exists a bounded linear operator S : R(T) → X for which ST = I on D(T) and R(T) = X, such that S = T−1 is necessarily uniquely determined and linear; the boundedness of S means that T must be bounded below, in the sense that there is M > 0 for which ‖Tx‖ ≥ M‖x‖ for all x ∈ D(T)
The matrix Λ is used for obtaining some new sequence spaces by its domain from the classical sequence spaces, it is not considered for determining the spectrum or fine spectrum acting as a linear operator on any of the classical sequence spaces c0, c, or lp
Summary
Let X and Y be Banach spaces, and let T : X → Y be a bounded linear operator. By B(X), we denote the set of all bounded linear operators on X into itself. If X is any Banach space and T ∈ B(X) the adjoint T∗ of T is a bounded linear operator on the dual X∗ of X defined by (T∗f)(x) = f(Tx) for all f ∈ X∗ and x ∈ X. The spectrum σ(T, X) consists of those α ∈ C, the complex field, for which Tα is not invertible, and the resolvent is the mapping from the complement σ(T, X) of the spectrum into the algebra of bounded linear operators on X defined by α → Tα−1
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