Bol. Soc. Paran. Mat. (3s.) v. ???? (??) : 1–10. c SPM –ISSN-2175-1188 on line ISSN-0037-8712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.44219 Subdivisions of the Spectra for D(r, 0, s, 0,t) Operator on Certain Sequence Spaces Avinoy Paul and Binod Chandra Tripathy abstract: In this paper we have examined the approximate point spectrum, defect spectrum and compres- sion spectrum of the operator D(r, 0, s, 0,t) on the sequence spaces c 0 , c and bvp(1 <p< ∞). Key Words: Fine spectrum, Approximate point spectrum, Defect spectrum, Compression spectrum. Contents 1 Preliminaries and Definition 1 2 Subdivisions of the spectrum 2 2.1 The point spectrum, continuous spectrum and residual spectrum .............. 2 2.2 The approximate point spectrum, defect spectrum and compression spectrum ...... 2 2.3 Goldberg’s classification of spectrum .............................. 3 3 Subdivisions of the spectrum of D(r, 0, s, 0,t) over c 0 5 4 Subdivisions of the spectrum of D(r, 0, s, 0,t) over c 6 5 Subdivisions of the spectrum of D(r, 0, s, 0,t) on (1 <p< ∞) 7 6 Subdivisions of the spectrum of D(r, 0, s, 0,t) on bv p (1 <p< ∞) 8 1. Preliminaries and Definition Let X and Y be Banach spaces and T : X → Y be a bounded linear operator. The set of all bounded linear operators on X into itself is denoted by B(X ). The adjoint T ∗ : X ∗ → X ∗ of T is defined by (T ∗ Φ)(x) = Φ(Tx) for all Φ ∈ X ∗ and x ∈ X . Clearly, T ∗ is a bounded linear operator on the dual space X ∗ . Let T : D(T ) → X a linear operator, defined on D(T ) ⊆ X , where D(T ) denote the domain of T and X is a complex normed linear space. For T ∈ B(X ) we associate a complex number α with the operator (T − αI ) denoted by T α defined on the same domain D(T ), where I is the identity operator. The inverse (T − αI ) −1 , denoted by T −1 α is known as the resolvent operator of T . Many properties of T α and T −1 α depend on α and spectral theory is concerned with those properties. We are interested in the set of all α in the complex plane such that T −1 α exists. Boundedness of T −1 α is another essential property. We also determine α ′ s for which the domain of T −1 α is dense in X . A regular value is a complex number α of T such that (R 1 )T −1 α exists, (R 2 )T −1 α is bounded and (R 3 )T −1 α is defined on a set which is dense in X . The resolvent set of T is the of all such regular values α of T , denoted by ρ(T,X ). Its complement is given by C \ ρ(T,X ) in the complex plane C is called the spectrum of T , denoted by σ(T,X ). Thus the spectrum σ(T,X ) consist of those values of α ∈ C, for which T α is not invertible. 2010 Mathematics Subject Classification: 40H05, 40C99, 46A35, 47A10. Submitted August 22, 2018. Published November 17, 2018 1 Typeset by B S P M style. c  Soc. Paran. de Mat.