Compact Difference Index in Fractional Spaces

• Compact Difference Index is a quantitative measure derived via the Hausdorff measure to assess how close an operator is to compactness in fractional difference sequence spaces.
• It utilizes transformed matrix entries from the fractional difference operator Δ^α to compute operator norms and establish precise criteria for compactness.
• Its application unifies analysis of both integer and noninteger order operators, facilitating rigorous classification and norm computation in sequence spaces.

The compact difference index quantifies the degree of noncompactness exhibited by operators acting between fractional difference sequence spaces. Defined via the Hausdorff measure of noncompactness, the compact difference index, denoted κ(A), characterizes the proximity of a given bounded linear operator to compactness within the context of sequence spaces generated by fractional difference operators. This metric arises naturally in the study of operators on the matrix domains of classical sequence spaces, notably those associated with the fractional difference operator Δ^α defined by the generalized binomial sum. The compact difference index, and the attendant criteria for compactness, provide a sharp analytic tool for operator classification, norm computation, and the general theory of difference sequence spaces of both integer and noninteger order (Özger, 2018).

1. Fractional Difference Sequence Spaces and Operators

Let ω denote the vector space of all complex sequences, with distinguished subspaces: c₀ (null sequences), c (convergent sequences), and ℓ_∞ (bounded sequences). The fractional difference operator Δ^α of order α ∈ ℝ acts on sequences via

$$(\Delta^\alpha x)_k = \sum_{i=0}^\infty (-1)^i{\alpha \choose i}x_{k-i},$$

where ${\alpha \choose i}$ is the fractional binomial coefficient, defined by

$${\alpha \choose i} = \frac{\Gamma(\alpha+1)}{i!\,\Gamma(\alpha-i+1)}$$

and Γ denotes the gamma function. For integer α = m ∈ ℕ, Δ^m reduces to the classical m-th forward difference operator.

Given a triangle matrix T, the matrix domain $X_T$ of a BK-space $X \subset \omega$ is defined as $X_T = \{x \in \omega:T x \in X\}$ with norm $\|x\|_{X_T} = \|T x\|_X$. Typical fractional difference sequence spaces include:

• $$c_0(\Delta^\alpha) = \{ x \in \omega : \Delta^\alpha x \in c_0 \}$$, normed by sup-norm,
• $$c(\Delta^\alpha) = \{ x \in \omega : \Delta^\alpha x \in c \}$$,
• $$\ell_\infty(\Delta^\alpha) = \{ x \in \omega : \Delta^\alpha x \in \ell_\infty \}.$$

A bounded linear operator $L_A$ between such spaces is induced by a matrix $A = (a_{n,k})$, acting by $$L_A x = A x = (\sum_{k=0}^\infty a_{n,k}x_k)_{n=0}^\infty$$ for all $x$ with convergent row-sums.

2. Associated Matrices and Transforms

To analyze operator norms and compactness properties, one introduces the associated matrix $\widehat{A}$:

$$\widehat{A}_{n,k} = \sum_{j=k}^\infty (-1)^{j-k} {\alpha \choose j-k} a_{n,j}$$

representing the action of the formal inverse fractional difference transform (i.e., the -α order).

This transform is essential, since norm and noncompactness estimates are expressed succinctly in terms of the entries of $\widehat{A}$. For targets $Y$ in ${c_0, c, \ell_\infty}$, these entries govern operator bounds and Hausdorff measures of noncompactness.

3. Operator Norms and the Hausdorff Measure of Noncompactness

The operator norm of $L_A:X \rightarrow Y$ is determined by the associated matrix $\widehat{A}$ via:

• For $Y = c_0$ or $\ell_\infty$,

$$\|L_A\| = \max_{n \ge 0} \sup_{k \ge 0} |\widehat{A}_{n,k}|.$$

• For $Y = c$ and limiting row $\alpha_k = \lim_{n \to \infty} \widehat{A}_{n,k}$,

$$\|L_A\| = \max_{n \ge 0} \left( \sup_k |\widehat{A}_{n,k} - \alpha_k| + |\alpha_n| \right).$$

• For $Y = \ell_1$, norm estimates involve finite sums of $\sup_k |\widehat{A}_{n,k}|$.

The Hausdorff measure of noncompactness $\chi(L_A)$ for $L_A$ is defined as the infimum of ε > 0 such that the image of the unit ball under $L_A$ can be covered by finitely many balls of radius ε. This measure vanishes if and only if $L_A$ is compact.

4. Definition and Calculation of the Compact Difference Index

The compact difference index, denoted $\kappa(A)$, is

$\kappa(A) := \chi(L_A)$

and quantifies the "distance to compactness" for $L_A$ between fractional difference sequence spaces. Sometimes, the relative index $\chi(L_A)/\|L_A\|$ is also considered.

To compute $\kappa(A)$, one proceeds by:

• Computing the transformed entries $\widehat{A}_{n,k}$ as above.
• Evaluating the tail supremum:
  • For $c_0$ or $\ell_\infty$ targets:

  $$\kappa(A) = \lim_{r \to \infty} \sup_{n > r} \sup_k |\widehat{A}_{n,k}|.$$ - For $c$ targets (with limits $\alpha_k$ defined as above):

  $$\kappa(A) = \lim_{r \to \infty} \sup_{n > r} \sup_k |\widehat{A}_{n,k} - \alpha_k|.$$ - For source $c(\Delta^\alpha)$ with $Y \in \{c_0, c, \ell_\infty\}$, one instead considers the vector supremum distance to a limiting row $B$.

A summary of the compactness conditions is provided in the following table:

Map type	Compactness criterion	$\kappa(A)$ formula
$\ell_\infty(\Delta^\alpha) \to c_0,\ell_\infty$	$\lim_{n \to \infty} \sup_k |\widehat{A}_{n,k}| = 0$	$$\lim_{r \to \infty} \sup_{n > r} \sup_k |\widehat{A}_{n,k}|$$
$\ell_\infty(\Delta^\alpha) \to c$	$$\lim_{n \to \infty} \sup_k |\widehat{A}_{n,k} - \alpha_k| = 0$$	$$\lim_{r \to \infty} \sup_{n > r} \sup_k |\widehat{A}_{n,k} - \alpha_k|$$
$c(\Delta^\alpha) \to c_0,c,\ell_\infty$	$$\lim_{n \to \infty} \|\widehat{A}_n - B\|_\infty = 0$$	$$\lim_{r \to \infty} \sup_{n > r}\|\widehat{A}_n - B\|_\infty$$

For $\ell_1$ targets, $\kappa(A)$ is given by a two-sided finite supremum sum estimate.

5. Criteria for Compactness and Relation to the Compact Difference Index

Compactness of $L_A$ is characterized by κ(A) = 0. Specifically:

• For $\ell_\infty(\Delta^\alpha) \to c_0$ and $\ell_\infty$, compactness is equivalent to the vanishing of the tail suprema of $\widehat{A}_{n,k}$.
• For mappings into $c$, the centered suprema $|\widehat{A}_{n,k} - \alpha_k|$ must vanish as $n \to \infty$.
• For $c(\Delta^\alpha)$ sources, the convergence of rows $\widehat{A}_n$ to a limit $B$ in the sup-norm is required.
• For mappings into $\ell_1$, the compactness criterion is provided by the two-sided estimate in Theorem 3.6 (Özger, 2018).

These conditions generalize the well-known integer-order difference case, extending sharp analytic criteria to arbitrary real order difference operators.

6. Illustrative Examples and Theoretical Remarks

Although numerical examples are not detailed in the referenced work, a prototypical application involves banded matrices, such as $a_{n,n}=1$, $a_{n,n-1}=-1/2$, and zeros elsewhere. The calculation of $\widehat{A}_{n,k}$ via binomial expansions allows determination of compactness, with κ(A) = 0 indicating the operator is compact and κ(A) > 0 otherwise.

The theoretical apparatus employs canonical Schauder bases, operator tail projections, and the reduction $A = \widehat{A}\Delta^\alpha$ to control the large-index contributions. The compact difference index therefore measures the degree to which the operator images of the unit ball are "eventually small-in-tail".

7. Connection to Integer-Order Difference Operators and Extensions

For α ∈ ℕ, $\Delta^m$ becomes the standard difference operator, and the fractional spaces $X(\Delta^m)$ coincide with classical difference sequence spaces of integer order. The associated compactness and norm formulas recover established results in the literature for the integer-order case. The binomial transform formulas for fractional order α yield more general results, applicable to a wider range of operators and spaces.

The compact difference index furnishes an explicit, quantitative measure of noncompactness for operators on fractional-difference sequence spaces, and serves as a bridge between concrete operator theory, sequence space generalizations, and the spectral theory of infinite matrices (Özger, 2018).