Fine Spectrum of the Generalized Difference Operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1">mml:miB<mml:mo stretchy="false">(mml:mirmml:mo,mml:mis<mml:mo stretchy="false">) over the Class of Converg... (original) (raw)

On the fine spectrum of the generalized difference operator over the sequence spaces and

Nonlinear Analysis: Theory, Methods & Applications, 2008

We determine the fine spectrum of the generalized difference operator B(r,s) defined by a band matrix over the sequence spaces c 0 and c, and derive a Mercerian theorem. This generalizes our earlier work (2004) for the difference operator ∆, and includes as other special cases the right shift and the Zweier matrices.

Some Spectral Aspects of the Operator over the Sequence Spaces and

Chinese Journal of Mathematics, 2013

The main idea of the present paper is to compute the spectrum and the fine spectrum of the generalized difference operator over the sequence spaces . The operator denotes a triangular sequential band matrix defined by with for , where or , ; the set nonnegative integers and is either a constant or strictly decreasing sequence of positive real numbers satisfying certain conditions. Finally, we obtain the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of the operator over the sequence spaces and . These results are more general and comprehensive than the spectrum of the difference operators , , , , and and include some other special cases such as the spectrum of the operators , , and over the sequence spaces or .

On the fine spectrum of the operator B (r, s, t) over the sequence spaces ℓp and bvp,(1< p<∞)

Appl. Math. Inf. Sci, 2011

The purpose of this paper is threefold: first to mainly review several recent results concerning the fine spectrum of the operator ∆v over the sequence spaces c and lp, where 1 < p < ∞; second to provide some new results concerning the residual spectrum and the continuous spectrum of the operator ∆ v over the sequence spaces c and l p ; and third to modify the definition of the operator ∆v and to determine the fine spectrum of the modified operator over the sequence spaces c and lp, where 1 < p < ∞. Also, it may be helpful to provide some comments and examples to support our results.

Subdivisions of the Spectra for D ( r , 0 , s , 0 , t ) Operator on Certain Sequence Spaces

2019

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 Tα defined on the same domain D(T ), where I is the identity operator. The inverse (T − αI), denoted by T α is known as the resolvent operator of T . Many properties of Tα and T 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 α exists. Boundedness of T −1 α is another essential property. We also determine αs for which the domain of T α is dense in X . A regular value is a complex numbe...