A-Compactness by Carl & Stephani

• The paper establishes an operator-ideal framework that generalizes compactness to include sets, linear operators, polynomials, and holomorphic mappings.
• It demonstrates the equivalence between operator-level and polynomial-level A-compactness using local-to-global principles and explicit counterexamples.
• The study introduces the precise A-compact radius, showing how Banach operator ideals extend classical notions and inform functional analysis.

The notion of $\mathcal{A}$-compactness, introduced by Carl and Stephani, provides a unifying operator-ideal-theoretic framework for studying compactness properties of sets, linear operators, polynomials, and holomorphic mappings between Banach spaces. By fixing a Banach operator ideal $\mathcal{A}$ (such as the ideals of compact, weakly compact, or $p$-nuclear operators), one investigates mappings whose local images are relatively $\mathcal{A}$-compact, extending classical compactness results to a more general ideal context. The advances by Turco (Turco, 2015) clarify the transfer, local, and structural properties of $\mathcal{A}$-compact sets and mappings, unveil the exact radius of $\mathcal{A}$-compact convergence, and demonstrate the sharpness of these descriptions through explicit counterexamples.

1. Banach Operator Ideals and Relatively $\mathcal{A}$-Compact Sets

A Banach operator ideal $\mathcal{A}$ is a subclass of continuous linear operators between Banach spaces endowed with an ideal norm $\|\cdot\|_{\mathcal{A}}$ satisfying an invariance property under composition: $$T\in\mathcal{L}(Y;Z),\, R\in\mathcal{A}(X;Y),\, S\in\mathcal{L}(W;X) \implies T\circ R\circ S\in\mathcal{A}(W;Z),\, \|T\circ R\circ S\|_{\mathcal{A}} \le \|T\|\|R\|_{\mathcal{A}}\|S\|.$$ Classical examples include the compact operators $\mathcal{K}$, weakly compact operators $\mathcal{W}$, and $p$-nuclear operators $\mathcal{N}_p$ for $1\le p<\infty$.

A set $K\subset X$ is called relatively $\mathcal{A}$-compact (Carl–Stephani) if there exist a Banach space $Z$, compact set $M\subset Z$, and $T\in\mathcal{A}(Z;X)$ with $K\subset T(M)$. The $\mathcal{A}$-measure of $K$ is

$$m_{\mathcal{A}}(K,X) = \inf \left\{ \|T\|_{\mathcal{A}} : K\subset T(M),\, M\subset B_Z\text{ compact} \right\},$$

with $m_{\mathcal{A}}(K,X)=+\infty$ if $K$ is not relatively $\mathcal{A}$-compact.

2. $\mathcal{A}$-Compact Operators: Mapping-Level Compactness

A bounded linear operator $T:X\rightarrow Y$ is $\mathcal{A}$-compact if $T(B_X)$ is relatively $\mathcal{A}$-compact in $Y$, and the collection of all such operators is $\mathcal{K}_{\mathcal{A}}(X;Y)$. This is equivalent to

$$T\in\mathcal{K}_{\mathcal{A}}(X;Y) \iff m_{\mathcal{A}}\left(T(B_X),Y\right)<\infty,$$

and $\mathcal{K}_{\mathcal{A}}$ is equipped with the norm

$$\|T\|_{\mathcal{K}_{\mathcal{A}}} = m_{\mathcal{A}}(T(B_X), Y).$$

The class $\mathcal{K}_{\mathcal{A}}$ itself forms a Banach operator ideal, and it inherits many permanence properties from $\mathcal{K}$ and $\mathcal{N}_p$, notably surjectivity.

3. $\mathcal{A}$-Compact Polynomials: Local-to-Global Principle

For $P:X\rightarrow Y$ a continuous $n$-homogeneous polynomial, $P$ is $\mathcal{A}$-compact if $P(B_X)$ is relatively $\mathcal{A}$-compact in $Y$. A pivotal result shows equivalence between polynomial-level and operator-level $\mathcal{A}$-compactness via linearization: $$P\in\mathrm{Pol}^n_{\mathcal{K}_{\mathcal{A}}}(X;Y) \iff L_P \in \mathcal{K}_{\mathcal{A}}(\widehat{\otimes}^n_{\pi,s}X; Y),$$ where $L_P$ is the linearization of $P$ and $\widehat{\otimes}^n_{\pi,s}X$ is the $n$-fold symmetric projective tensor power.

A central theorem (Proposition 2.6 in (Turco, 2015)) establishes the local determination of $\mathcal{A}$-compactness:

• $P$ is $\mathcal{A}$-compact iff there exists $x_0\in X$ and $\epsilon>0$ so that $P(x_0+\epsilon B_X)$ is relatively $\mathcal{A}$-compact, which is also equivalent to compactness at the origin.

This property relies on the polarization and homogeneity of $P$, allowing reduction from a neighborhood to the full unit ball.

4. $\mathcal{A}$-Compact Holomorphic Mappings and Radius of Convergence

Given $f:X\rightarrow Y$ holomorphic, $f$ is $\mathcal{A}$-compact at $x_0$ if $f(x_0+\epsilon B_X)$ is relatively $\mathcal{A}$-compact in $Y$. The $\mathcal{A}$-compact radius of convergence is defined as

$$r_{\mathcal{K}_{\mathcal{A}}}(f, x_0) = \frac{1}{\displaystyle \limsup_{n\to\infty}\|P_n f(x_0)\|_{\mathcal{K}_{\mathcal{A}}}^{1/n}},$$

where $P_n f(x_0)$ is the $n$-homogeneous Taylor coefficient of $f$ at $x_0$.

A characterization theorem (Proposition 3.4 in (Turco, 2015)) asserts the equivalence:

• $f$ is $\mathcal{A}$-compact at $x_0$ iff every Taylor term $$P_n f(x_0)\in \mathrm{Pol}^n_{\mathcal{K}_{\mathcal{A}}}(X;Y)$$ and $r_{\mathcal{K}_{\mathcal{A}}}(f, x_0)>0$.

Uniform convergence and diagonal-type selection lemmas for relatively $\mathcal{A}$-compact sets underlie this result. Once $f$ is $\mathcal{A}$-compact at $x_0$, compactness extends to the shifted ball $x_0 + r_{\mathcal{K}_{\mathcal{A}}}(f, x_0)B_X$.

5. Counterexamples and Sharpness of the Radius Criterion

Under the existence of some relatively $\mathcal{B}$-compact set not $\mathcal{A}$-compact, Turco presents two counterexample families:

• There exists $f:\ell_1\rightarrow X$ holomorphic, $\mathcal{B}$-compact and $\mathcal{A}$-compact at $0$ with $r_{\mathcal{K}_{\mathcal{A}}}(f,0)=1$, but $f$ fails to be $\mathcal{A}$-compact at $e_1$. This demonstrates the precise role of the local $\mathcal{A}$-compact radius in restricting where compactness holds.
• There exists $f\in H(\ell_1; X)$ such that every Taylor polynomial (at every point) lies in $\mathrm{Pol}^n_{\mathcal{K}_{\mathcal{A}}}$ but $f$ is nowhere $\mathcal{A}$-compact. The characterization theorem's radius condition is thus strict and not relaxable.

6. Unifying Ideal-Theoretic Framework and Applications

$\mathcal{A}$-compactness subsumes classical notions of compact, weakly compact, $p$-compact, and nuclear mappings under the formalism of Banach operator ideals, providing a flexible language for generalizing results about compactness in functional analysis. The extension of finite-order and local-to-global principles—known for compact or weakly compact mappings—to $\mathcal{A}$-compact mappings, polynomials, and holomorphic functions, constitutes the principal advance in Turco’s work (Turco, 2015). The identification and sharp characterization through the $\mathcal{A}$-compact radius, together with explicit counterexamples, elucidate how ideal properties propagate through multilinear and holomorphic structures. This suggests further study in operator theory and functional analysis may profitably employ the Carl–Stephani notion of relative $\mathcal{A}$-compactness in elaborating spectral, summing, and factorization properties across general classes of maps.