Characterization Of Compact Operators Research Articles

Compact operators on Cesàro sequence spaces and norms of Cesàro operators

This paper deals with the characterization of compact operators on Ces?ro sequence spaces as an application of Hausdorff measure of noncompactness. Further, the norms of Ces?ro operators on certain spaces are investigated.

Domain of q-Cesàro matrix in Hahn sequence space hd and the space bv of the sequences of bounded variation

Let hd = n f = (fk) ? ? : P k dk| fk ? fk+1| < ? o ? c0, where d = (dk) is an unbounded and monotonic increasing sequence of positive reals. We study the matrix domains hd(Cq) = (hd)Cq and bv(Cq)=(bv)Cq, where Cq is the q-Ces?ro matrix, 0 < q < 1. Apart from the inclusion relations and Schauder basis, we compute ?-, ?- and ?-duals of the spaces hd(Cq) and bv(Cq). We state and prove theorems concerning characterization of matrix classes from the spaces hd(Cq) and bv(Cq) to any one of the space ??, c, c0 or ?1. Finally, we obtain certain identities concerning characterization of compact operators using Hausdorff measure of non-compactness in the space hd(Cq).

Characterizations of compact operators on ℓp−type fractional sets of sequences

Abstract Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some ℓp−type fractional difference sets via the gamma function. Although, we characterize compactness conditions on those sets using the main key of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is ℓ∞. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.

Two methods for the characterization of compact operators between BK spaces

We establish some identities or inequalities for the Hausdorff measure of noncompactness for operators $L\in\mathcal{B}(X,Y)$ when $X=\ell_{p}$ $(p \in [1, \infty))$ and $Y=c$; $X=\ell_{p}$ $(p \in (1, \infty))$ and $Y=\ell_{\infty}$; $X=bv_{0}$ and $Y=c$; $X=c_{0}(\Delta),c(\Delta),\ell_{\infty}(\Delta)$ and $Y=\ell_{\infty}$. These identities and estimates are used to establish necessary and sufficient conditions for the operators to be compact. Furthermore, we apply a result by Sargent to establish necessary and sufficient conditions for operators in $\mathcal{B}(bv_{0},\ell_{\infty})$ and $\mathcal{B}(\ell_{1},Y)$ to be compact, where $Y=w_{\infty},v_{\infty}, [c]_{\infty}$.

Characterization of compact operators between certain BK spaces

We give a survey of the known results concerning the sets c0(Λ), c(Λ) and c∞(Λ) including their basic topological properties, their first and second dual spaces, and the characterizations of matrix transformations from them into the spaces l∞, c and c0. Furthermore, we establish some new results such as the representations of the general bounded linear operators from c(Λ) into the spaces l∞, c and c0, and estimates for their Hausdorff measures of noncompactness. Finally, we apply our results to characterize some classes of compact operators on c0(Λ), c(Λ) and c∞(Λ). We also generalize a classical result by Cohen and Dunford which states that a regular matrix operator cannot be compact.

A Characterization of Compact Operators Via the Non-Connectedness of the Attractors of a Family of IFSs

In this paper we present a result which establishes a connection between the theory of compact operators and the theory of iterated function systems. For a Banach space \(X\), \(S\) and \(T\) bounded linear operators from \(X\) to \(X\) such that \(\Vert S\Vert , \Vert T\Vert <1\) and \(w\in X\), let us consider the IFS \(\mathcal S _{w}=(X,f_{1},f_{2})\), where \(f_{1},f_{2}:X\rightarrow X\) are given by \(f_{1}(x)=S(x)\) and \(f_{2}(x)=T(x)+w\), for all \(x\in X\). On one hand we prove that if the operator \(S\) is compact, then there exists a family \((K_{n})_{n\in \mathbb N }\) of compact subsets of \(X\) such that \(A_{\mathcal S _{w}}\) is not connected, for all \(w\in X-\bigcup _{n\in \mathbb N } K_{n}\). On the other hand we prove that if \(H\) is an infinite dimensional Hilbert space, then a bounded linear operator \(S:H\rightarrow H\) having the property that \(\Vert S\Vert <1\) is compact provided that for every bounded linear operator \(T:H\rightarrow H\) such that \(\Vert T\Vert <1\) there exists a sequence \((K_{T,n})_{n}\) of compact subsets of \(H\) such that \(A_{\mathcal S _{w}}\) is not connected for all \(w\in H-\bigcup _{n}K_{T,n}\). Consequently, given an infinite dimensional Hilbert space \(H\), there exists a complete characterization of the compactness of an operator \(S:H\rightarrow H\) by means of the non-connectedness of the attractors of a family of IFSs related to the given operator. Finally we present three examples illustrating our results.

Infinite System of Differential Equations in Some BK Spaces

The first measure of noncompactness was defined by Kuratowski in 1930 and later the Hausdorff measure of noncompactness was introduced in 1957 by Goldenštein et al. These measures of noncompactness have various applications in several areas of analysis, for example, in operator theory, fixed point theory, and in differential and integral equations. In particular, the Hausdorff measure of noncompactness has been extensively used in the characterizations of compact operators between the infinite‐dimensional Banach spaces. In this paper, we present a brief survey on the applications of measures of noncompactness to the theory of infinite system of differential equations in some BK spaces c0, c, ℓp(1 ≤ p ≤ ∞) and n(ϕ).

Characterizations of compact operators on some Euler spaces of difference sequences of order m

In the past, several authors studied spaces of m-th order difference sequences, among them, H.Polat and F.Başar ([17]) defined the Euler spaces of m-th order difference sequences e 0 r ( Δ ( m ) ) , e c r ( Δ ( m ) ) and e ∞ r ( Δ ( m ) ) and characterized some classes of matrix transformations on them. In our paper, we add a new supplementary aspect to their research by characterizing classes of compact operators on those spaces. For that purpose, the spaces are treated as the matrix domains of a triangle in the classical sequence spaces c 0, c and l ∞. The main tool for our characterizations is the Hausdorff measure of noncompactness.

Compact Toeplitz operators on the Segal–Bargmann space

In this note, we show that Miao and Zheng's characterization of compact operators on the Bergman space of the unit disk that are finite sums of finite products of Toeplitz operators (with each one of the symbols belonging to BT) also holds for the Segal–Bargmann space of C n .

Some characterizations of compact operators on Banach lattices

We study the compactness of the class of operators which are AM-compact and semi-compact on Banach lattices and as consequences, we obtain some characterizations of order continuous norms.

An elementary proof of a theorem of tacon

In this Note we give an elementary proof (based on a simple, seemingly new, characterization of compact operators) of a Theorem of Tacon [3].