2.1 Bounded to bounded case

The main definition of this work follows:

Definition 2.1 Let $\mathcal {I},\, \mathcal {J}$ be ideals on $\omega$ and fix $T \in \mathcal {L}(X,\,Y)$. Then a matrix $A=(A_{n,k})$ of linear operators in $\mathcal {L}(X,\,Y)$ is said to be $(\mathcal {I},\, \mathcal {J})$-regular with respect to $T$ if

\[ A \in (c^b(X,\mathcal{I}),c^b(Y,\mathcal{J})) \,\,\, \text{ and }\,\,\, \mathcal{J}\text{-}\lim A\boldsymbol{x}=T(\,\mathcal{I}\text{-}\lim \boldsymbol{x}) \,\text{ for all }\boldsymbol{x} \in c^b(X,\mathcal{I}). \]

If $T$ is the identity operator $I$ on $X$ (namely, $Ix=x$ for all $x \in X$), we simply say that $A$ is $(\mathcal {I},\,\mathcal {J})$-regular. Note that if $\mathcal {I}=\mathcal {J}=\mathrm {Fin}$, $X=Y=\mathbf {R}$, and $T=I$, then definition 2.1 corresponds to the ordinary regular matrices.

The following result, essentially due to Robinson [Reference Robinson38, Theorem VII], is the (unbounded) operator version of theorem 1.1, cf. also [Reference Maddox32, Theorem 1].

A variant for continuous linear operators between Fréchet spaces has been proved by Ramanujan in [Reference Ramanujan37], cf. also [Reference Li, Kang and Swartz30, Corollary 6].

Notice that (S3) can be rewritten as $\lim _n \sum _{k \in E}A_{n,k}x_k=0$ for all sequences $\boldsymbol {x} \in \ell _\infty (X)$ and $E \in \mathrm {Fin}$, which is also equivalent to $A \in (c_{00}(X),\, c_0(Y))$.

Remark 2.4 Another difference from the finite-dimensional case is that a $(\mathrm {Fin},\, \mathrm {Fin})$-regular matrix does not necessarily belong to $(\ell _\infty (X),\,\ell _\infty (Y))$. For, suppose that $X=Y=\ell _2$, and let $e_k$ be the $k$th unit vector of $X$ for each $k \in \omega$. Building on the above example, consider the matrix $A=(A_{n,k})$ of linear operators in $\mathcal {L}(\ell _2,\,\ell _2)$ such that $A=\mathrm {Id}+B$, where $\mathrm {Id}$ is the identity matrix and

\[ \forall n,k \in \omega, \forall x \in \ell_2, \quad B_{n,k}(x):= \begin{cases} (0,\ldots,0,x_{n+1},x_{n+2},\ldots)\,\, & \text{if }k=0;\\ - x_{n+k}e_{n+k} & \text{if }k>0. \end{cases} \]

Then $A$ satisfies conditions (S1)–(S3) with $T=I$ and $k_0=0$, hence by theorem 1.1 $A$ is a $(\mathrm {Fin},\, \mathrm {Fin})$-regular matrix. However, the sequence $\boldsymbol {x}:=(e_0,\,e_1,\,\ldots ) \in \ell _\infty (\ell _2)$ does not belong to $\mathrm {dom}(A)$, indeed $A_0\boldsymbol {x}=e_0-e_1-e_2-\cdots$ is not norm convergent in $\ell _2$. Therefore $A\notin (\ell _\infty (X),\,\ell _\infty (Y))$, cf. theorem 3.8 below.

Our first main result, which corresponds to the operator version of [Reference Connor and Leonetti11, Theorem 1.2], follows.

For the sake of clarity, condition (T4) means that $A_n(x,\,x,\,\ldots )$ is norm convergent for all $n \in \omega$ and $x \in X$ (which is weaker than (T3)) and, in addition, $\mathcal {J}\text {-}\lim _n A_n(x,\,x,\,\ldots )=Tx$ for all $x \in X$. Lastly, condition (T5) can be rephrased as: $A_n\boldsymbol {x}$ is norm convergent for all $n \in \omega$ and all bounded sequences $\boldsymbol {x}$ supported on $\mathcal {I}$ and, for such sequences, $\mathcal {J}\text {-}\lim A\boldsymbol {x}=0$.

Even if theorem 2.5 may look quite complicated, the reader should keep in mind that it deals with (possibly unbounded) linear operators and general ideal/filter convergence. We are going to see that, in some special circumstances, it may be considerably simplified because either some of conditions (T1)–(T5) are automatically satisfied or the latter ones collapse to simpler properties (in particular, recovering the classical ones). Several related results may be found in the literature in the case $X=Y$ equal to $\mathbf {R}$ or $\mathbf {C}$, $T$ equals to the identity operator $I$ or the zero operator, and $\mathcal {I},\,\mathcal {J}$ being certain $F_{\sigma \delta }$-ideals (where ideals are regarded as subsets of the Cantor space $\{0,\,1\}^\omega$), see e.g. [Reference Connor and Gumus10, Reference Demirci, Khan and Orhan13, Reference Drewnowski14, Reference Kolk24, Reference Tripathy43].

We remark also that, if $T$ is not bounded, then an $(\mathcal {I},\,\mathcal {J})$-regular matrix $A$ with respect to $T$ may not exist: indeed, if each $A_{n,k}$ is bounded and $\mathcal {J}=\mathrm {Fin}$, condition (T4) and the Banach–Steinhaus theorem imply that $T$ is necessarily bounded.

In the case that $T=0$, we obtain the following immediate consequence, cf. also corollary 2.12 below for the finite-dimensional case with $\mathcal {I}$ maximal.

It will be useful to define also the following properties:

1. (T3♮) $\lim _k \|A_{n,\ge k}\|=0$ for all $n\in \omega$;

2. (T6♭) $\mathcal {J}\text {-}\lim _n \|A_{n,k}\|=0$ for all $k\in \omega$.

It is clear that (S4) corresponds to (T6$^\flat$) in the case $\mathcal {J}=\mathrm {Fin}$. Some implications between the above-mentioned conditions are collected below.

It is immediate to check that theorem 2.2 comes as a corollary, putting together theorem 2.5 and proposition 2.8(i) and (ii).

However, the usefulness of a characterization of $(\mathcal {I},\, \mathcal {J})$-regular matrices with respect to $T$ comes from the practical easiness to check whether conditions (T1)–(T5) hold together. Taking into account the implications given in proposition 2.8, it is evident that (T5) is the most demanding in this direction. Hence it makes sense to search for sufficient conditions which allow us to simplify it. In the same spirit of [Reference Connor and Leonetti11, Theorem 1.3], which studies the case $X=Y=\mathbf {R}$ and $T=I$, we obtain characterizations of such matrices which avoid condition (T5). We need the new and much “easier” condition:

1. (T6) $\mathcal {J}\text {-}\lim _n \|A_{n,E}\|=0$ for all $E \in \mathcal {I}$.

Directly by the definition of group norm, it is clear that (T6) implies (T5) (and also (T6$^\flat$)). This means that we are allowed to replace (T5) with the stronger condition (T6) provided that the latter is satisfied for matrices $A$ which are $(\mathcal {I},\, \mathcal {J})$-regular with respect to $T$, possibly under some additional constraints.

On a similar direction, recall that, if $X,\,Y$ are vector lattices, then a linear operator $T \in \mathcal {L}(X,\,Y)$ is said to be positive if $Tx\ge 0$ whenever $x\ge 0$. In addition, a Banach space $V$ is called an AM-space if $V$ is also a vector lattice such that $0\le x\le y$ implies $\|x\| \le \|y\|$, and $\|x \vee y\|=\max \{\|x\|,\,\|y\|\}$ for all $x,\,y \ge 0$; we say that $e \in V$ is an order unit if, for all $x \in V$ there exists $n \in \omega$ such that $-ne \le x\le ne$. Accordingly, if $V\neq \{0\}$, then necessarily $e>0$. Examples of AM-spaces with order units include $\ell _\infty$ and $C(K)$ spaces, for some compact Hausdorff space $K$. We refer to [Reference Aliprantis and Burkinshaw2, Reference Aliprantis and Burkinshaw3] for the underlying theory on vector lattices.

In the finite-dimensional case, everything is simpler. Indeed, suppose that $X=\mathbf {R}^d$ and $Y=\mathbf {R}^m$, for some integers $d,\,m\ge 1$. Then each linear operator $A_{n,k}$ is represented by the real matrix $[\,a_{n,k}(i,\,j): 1\le i\le m,\, 1\le j\le d\,]$ and $T$ is represented by the real matrix $[\,t(i,\,j): 1\le i\le m,\, 1\le j\le d\,]$.

It is remarkable that the “easier” characterization with condition (F3) does not hold uniformly for all ideals $\mathcal {I},\, \mathcal {J}$: indeed, it has been proved in [Reference Connor and Leonetti11, Theorem 1.4] that, even in the simplest case $X=Y=\mathbf {R}$ and $T=I$, there exists a $(\mathcal {Z},\, \mathcal {Z})$-regular matrix which does not satisfy (F3), where $\mathcal {Z}$ is the asymptotic density zero ideal defined in (2.1). In addition, condition (F3) can be simplified if $T$ is the zero operator and $\mathcal {I}$ a maximal ideal:

This provides a generalization of [Reference Çakan and Altay8, Lemma 3.2] in the case $d=m=1$, $T=0$, and $\mathcal {J}$ equal to the countably generated ideal $\mathcal {I}_{\mathrm {P}}$ defined below in (2.3).

A similar result can be obtained if each $A_{n,k}$ is a multiple of a given linear operator:

In particular, under the hypotheses of corollary 2.13, if $A$ is $(\mathcal {I},\,\mathcal {J})$-regular with respect to a linear operator $T$, then $T$ is necessarily bounded.

In the following sections, we obtain the analogues of theorem 2.5 where we replace the bounded sequence spaces $c^b(X,\,\mathcal {I})$ and $c^b(Y,\,\mathcal {J})$ of definition 2.1 with their unbounded versions $c(X,\,\mathcal {I})$ and $c(Y,\,\mathcal {J})$.

2.2 Bounded to unbounded case

An ideal $\mathcal {J}$ on $\omega$ is said to be a rapid $^+$-ideal if, for every $S \in \mathcal {J}^+$ and $F \in \mathrm {Fin}^+$, there exists $S^\prime \subseteq S$ such that $S^\prime \in \mathcal {J}^+$ and $|S^\prime \cap [0,\,n]|\le |F \cap [0,\,n]|$ for all $n\in \omega$. Moreover, $\mathcal {J}$ is called a $P^+$-ideal if, for every decreasing sequence $(S_n)$ in $\mathcal {J}^+$, there exists $S \in \mathcal {J}^+$ such that $S\setminus S_n$ is finite for all $n \in \omega$. The class of rapid$^+$ and $P^+$-ideals have been studied also, e.g. in [Reference De Bondt and Vernaeve12, Reference García-Ferreira and Pino-Villela18, Reference Hrušák21, Reference Matet and Pawlikowski35]. The ideal $\mathcal {J}$ is said to be countably generated if there exist a sequence $(Q_j)$ of subsets of $\omega$ such that $S \in \mathcal {J}$ if and only if $S\subseteq \bigcup _{j \in F}Q_j$ for some $F \in \mathrm {Fin}$.

Moreover, an ideal $\mathcal {J}$ on $\omega$ is said to be selective if, for every decreasing sequence $(S_n)$ in $\mathcal {J}^+$, there exists $S \in \mathcal {J}^+$ such that $S\setminus [0,\,n] \subseteq S_n$ for all $n \in \omega$, see e.g. [Reference Todorcevic42, Definition 7.3]. (They were introduced by Mathias in [Reference Mathias36] under the name of happy families.) It is easy to see, directly from the definitions, that every selective ideal is a rapid$^+$ $P^+$-ideal. More precisely, it is known that an ideal $\mathcal {J}$ on $\omega$ is selective if and only if $\mathcal {J}$ is a $P^+$-ideal and, in addition, for every sequence $(F_n)$ of finite subsets of $\omega$ with $F:=\bigcup _n F_n \in \mathcal {J}^+$, there exists $S \subseteq F$ such that $S \in \mathcal {J}^+$ and $|S \cap F_n|\le 1$ for all $n \in \omega$, see e.g. [Reference Todorcevic42, Lemma 7.4]. We refer to [Reference Kwela and Zakrzewski26, Section 1] for a list of known examples of selective ideals. It is a folklore fact that every $F_\sigma$-ideal is a $P^+$-ideal, see e.g. [Reference Just and Krawczyk22]. However, the summable ideal $\mathcal {I}_{1/n}:=\{S\subseteq \omega : \sum _{n \in S}1/(n+1)<\infty \}$ is an $F_\sigma$-ideal which is not rapid$^+$, and $\mathcal {Z}$ is neither a rapid$^+$ nor $P^+$-ideal (hence, they are not selective). This does not mean that the topological complexity of selective ideals is low: indeed, under Martin’s axiom for countable posets, there exist uncountably many nonisomorphic maximal selective ideals (on the other hand, their existence is not provable in ZFC), see [Reference De Bondt and Vernaeve12, Section 5.1].

For the next results, we need a slightly stronger version of selectivity: an ideal $\mathcal {J}$ on $\omega$ is called strongly selective if, for every decreasing sequence $(S_n)$ in $\mathcal {J}^+$, there exists $S=\{x_n: n \in \omega \} \in \mathcal {J}^+$ such that $x_{n+1} \in S_{x_n}$ for all $n \in \omega$. (Here, we are using $(x_n: n \in \omega )$ for the canonical enumeration of $S$. We are not aware whether this naming has been used somewhere else.) Of course, every strongly selective ideal is also selective. Note that every countably generated ideal is strongly selective (hence also rapid$^+$ and $P^+$); in particular, $\mathrm {Fin}$ is a strongly selective ideal, cf. remark 2.16 below. More generally, it is easy to see that, if $\mathcal {J}$ is an ideal on $\omega$ with the Baire property, then $\mathcal {J}$ is selective if and only if it is strongly selective. Moreover, as pointed out by the referee, maximal strongly selective ideals coincide with maximal selective ideals, thanks to known characterizations of the latter ones (in particular, by the observations above, their existence can be shown under Martin’s axiom).

Lastly, we need the following weakening of condition (T1) (and they coincide if $\mathcal {J}=\mathrm {Fin}$):

1. (T1♭) There exists $J_0 \in \mathcal {J}^\star$ for which $\sup _{n \in J_0}\|A_{n,\ge k_0}\|<\infty$ and, for all $n \in \omega \setminus J_0$, there exists $f(n) \in \omega$ such that $\|A_{n,\ge f(n)}\|<\infty$.

This condition has been suggested by the example given in [Reference Connor and Leonetti11, Section 4].

With these premises, we state the analogue of theorem 2.5 for the unbounded codomain sequence spaces.

Theorem 2.14 Fix a linear operator $T \in \mathcal {L}(X,\,Y)$, where $X,\,Y$ are Banach spaces. Let also $\mathcal {I},\,\mathcal {J}$ be an ideal on $\omega$ such that $\mathcal {J}$ is a strongly selective ideal.

Then a matrix $A=(A_{n,k})$ of linear operators in $\mathcal {L}(X,\,Y)$ satisfies

(2.2)\begin{equation} A \in (c^b(X,\mathcal{I}),c(Y,\mathcal{J})) \quad \text{and}\quad \mathcal{J}\text{-}\lim A\boldsymbol{x}=T(\,\mathcal{I}\text{-}\lim \boldsymbol{x}) \,\text{ for all }\boldsymbol{x} \in c^b(X,\mathcal{I}). \end{equation}

if and only if there exists $k_0 \in \omega$ such that (T1$^\flat$), (T3), (T4), and (T5) hold.

In addition, if each $A_{n,k}$ is bounded, it is possible to choose $k_0=f(n)=0$ for all $n\in \omega \setminus J_0$.

During the proof, need an ideal version of the Banach–Steinhaus theorem which has been recently proved in [Reference De Bondt and Vernaeve12], see theorem 3.5 below. Interestingly, the latter result provides a characterization of rapid$^+$ $P^+$-ideals [Reference De Bondt and Vernaeve12, Theorem 5.1], which suggests that theorem 2.14 cannot be improved with the current techniques.

We remark that theorem 2.14 sheds light on the substantial difference between the classical Silverman–Toeplitz characterization stated in theorem 1.1 and its ‘ multidimensional” analogue proved by Robinson [Reference Robinson39] and Hamilton [Reference Hamilton19] for double sequences, namely, the weakening of (T1) to (T1$^\flat$). For, recall that a double sequence $(x_{m,n}: m,\,n \in \omega )$ has Pringsheim limit $\eta \in X$, shortened as $\mathrm {P}\text {-}\lim _{m,n}x_{m,n}=\eta$, if for all $\varepsilon >0$ there exists $k \in \omega$ such that $\|x_{m,n}-\eta \|<\varepsilon$ for all $m,\,n \ge k$. At this point, define the ideal:

(2.3)\begin{equation} \mathcal{I}_{\mathrm{P}}:=\left\{S\subseteq \omega: \sup\nolimits_{n \in S}\nu_2(n)<\infty\right\}, \end{equation}

where $\nu _2$ is the $2$-adic valution defined by $\nu _2(0):=0$ and $\nu _2(n):=\max \{k \in \omega : 2^k \text { divides }n\}$ if $n>0$. Note that the ideal $\mathcal {I}_{\mathrm {P}}$ is countably generated by the sequence of sets $(Q_t: t \in \omega )$, where $Q_t:=\left \{S\subseteq \omega : \sup \nolimits _{n \in S}\nu _2(n)=t\right \}$ for all $t \in \omega$. Hence, $\mathcal {I}_{\mathrm {P}}$ is a strongly selective ideal. Let also $h: \omega ^2\to \omega$ be an arbitrary bijection with the property that $h[\{(m,\,n) \in \omega ^2: \min \{m,\,n\}=k\}]=Q_k$ for all $k \in \omega$. Thus, we obtain

(2.4)\begin{equation} \mathrm{P}\text{-}\lim\nolimits_{m,n}x_{m,n}=\eta \quad \text{ if and only if }\quad \mathcal{I}_{\mathrm{P}}\text{-}\lim\nolimits_n x_{h^{{-}1}(n)}=\eta, \end{equation}

as it has been observed in [Reference Leonetti28, Section 4.2], cf. also [Reference Drewnowski14]. In other words, $\mathcal {I}_{\mathrm {P}}$ is an isomorphic copy on $\omega$ of the ideal on $\omega ^2$ generated by vertical lines and horizontal lines, cf. remark 2.16 below. Relying on equivalence (2.4), the classical definition of RH-regular matrix $A$ coincides with (2.2) in the case $X=Y=\mathbf {R}$, $T=I$, and $\mathcal {I}=\mathcal {J}=\mathcal {I}_{\mathrm {P}}$. With the same notations of corollary 2.11, we can state the following consequence in the finite-dimensional case.

It is clear that, if also $\mathcal {I}$ is countably generated by a sequence $(E_t: t \in \omega )$, which is the case of RH-regular matrices, then (R4) can be rewritten as:

1. (R4′ ) $\mathcal {J}\text {-}\lim _n \sum _{k\in E_t}\sum _{i,j}\left |a_{n,k}(i,\,j)\right |=0$ for all $t \in \omega$.

Another special instance of corollary 2.15 has been proved in [Reference Connor9, Theorem 5] for the case where $X=Y=\mathbf {R}$, $T=I$, $A$ is a RH-regular matrix with nonnegative real entries, $\mathcal {I}$ is a $P$-ideal (i.e. if $(S_n)$ is an increasing sequence in $\mathcal {I}$, there exists $S \in \mathcal {I}$ such that $S_n\setminus S \in \mathrm {Fin}$ for all $n$), and $\mathcal {J}=\mathrm {Fin}$.

Other consequences of theorem 2.14, in the same vein of the ones given in § 2.1, may be obtained here, and they are left to the reader.

Remark 2.16 An ideal $\mathcal {I}$ on $\omega$ is countably generated if and only if it is isomorphic to one of the following:

1. (i) $\mathrm {Fin}$;

2. (ii) $\mathrm {Fin}\times \emptyset :=\{S\subseteq \omega ^2: \exists n \in \omega,\, S\subseteq [0,\,n]\times \omega \}$;

3. (iii) $\mathrm {Fin}\oplus \mathcal {P}(\omega ):=\{S\subseteq \{0,\,1\}\times \omega : |S\cap (\{0\}\times \omega )|<\infty \}$.

(Recall that two ideals $\mathcal {I}_1$ and $\mathcal {I}_2$ on countable sets $H_1$ and $H_2$, respectively, are called isomorphic, written as $\mathcal {I}_1 \simeq \mathcal {I}_2$ if there exists a bijection $h: H_1\to H_2$ such that $h[S] \in \mathcal {I}_2$ if and only if $S \in \mathcal {I}_1$ for all $S\subseteq H_1$; accordingly, it is easy to see that $\mathcal {I}_{\mathrm {P}} \simeq \mathrm {Fin}\times \emptyset$, and that the ideals in (i)–(iii) are pairwise nonisomorphic.) This has been essentially proved in [Reference Farah15, Proposition 1.2.8], however the correct statement appears in [Reference Balcerzak, Głab and Swaczyna4, Section 2].

We include its simple proof for the sake of completeness. Suppose that $\mathcal {I}$ is countably generated by a partition $\{Q_j: j \in \omega \}$ of $\omega$, and define $J:=\{j \in \omega : Q_j \in \mathrm {Fin}\}$. If $J$ is empty then every $Q_j$ is infinite, hence $\mathcal {I}\simeq \mathrm {Fin}\times \emptyset$. If $J$ is nonempty finite then $Q_j$ is infinite for infinitely many $j\in \omega$, hence $\mathcal {I}\simeq \mathcal {J}$, where

\[ \mathcal{J}:=\mathrm{Fin}\oplus (\mathrm{Fin}\times \emptyset):=\{S\subseteq \omega \cup \omega^2: S\cap \omega \in \mathrm{Fin}, S\cap \omega^2 \in \mathrm{Fin}\times \emptyset\}. \]

However, $\mathcal {J}\simeq \mathrm {Fin}\times \emptyset$, with the witnessing bijection $h: \omega \cup \omega ^2\to \omega ^2$ defined by $h(a,\,b)=(a,\,b+1)$ and $h(a)=(a,\,0)$ for all $a,\,b \in \omega$. Hence, let us assume hereafter that $J$ is infinite. If $J^c$ is empty then $\mathcal {I}\simeq \mathrm {Fin}$. If $J^c$ is nonempty finite then $\mathcal {I}\simeq \mathrm {Fin}\oplus \mathcal {P}(\omega )$. Lastly, if also $J^c$ is infinite, then $\mathcal {I}\simeq \mathcal {J}\simeq \mathrm {Fin}\times \emptyset$.

2.3 Unbounded to bounded case

In this section, we may assume that $\mathcal {I}\neq \mathrm {Fin}$, otherwise we go back in the previous cases. Differently from the other results, we are going to show that, quite often, there are no matrices $A$ which satisfy

(2.5)\begin{equation} A \in (c(X,\mathcal{I}),c^b(Y,\mathcal{J})) \quad \text{and}\quad \mathcal{J}\text{-}\lim A\boldsymbol{x}=T(\,\mathcal{I}\text{-}\lim \boldsymbol{x}) \,\text{ for all }\boldsymbol{x} \in c(X,\mathcal{I}), \end{equation}

unless $T$ is the zero operator. To this aim, recall that an ideal $\mathcal {I}$ is said to be tall if, for every infinite set $S\subseteq \omega$, there exists an infinite subset $S^\prime \subseteq S$ which belongs to $\mathcal {I}$ (note that countably generated ideals, hence also $\mathrm {Fin}$, are not tall; for a necessary condition in the case of countably generated ideals $\mathcal {J}$ and arbitrary $\mathcal {I}$, cf. remark 2.23 below).

Of course, if $T=0$, then the zero matrix $A$ (namely, the matrix with $A_{n,k}=0$ for all $n,\,k$) satisfies (2.5). However, this is essentially the unique possibility:

As it will turn out, condition (B1) is satisfied also for all matrices in the larger class $(c_{00}(X,\,\mathcal {I}),\, \ell _\infty (Y))$, provided that $\mathcal {I}$ is tall.

2.4 Unbounded to unbounded case

In this last section, we study the analogue condition for a matrix $A=(A_{n,k})$ of linear operators in $\mathcal {L}(X,\,Y)$ to satisfy

(2.6)\begin{equation} A \in (c(X,\mathcal{I}),c(Y,\mathcal{J})) \quad \text{and}\quad \mathcal{J}\text{-}\lim A\boldsymbol{x}=T(\,\mathcal{I}\text{-}\lim \boldsymbol{x}) \,\text{ for all }\boldsymbol{x} \in c(X,\mathcal{I}). \end{equation}

In the next results, we need a further weakening of (T1$^\flat$) and stronger versions of conditions (T3) and (T5), that is,

1. (T1^♭♭) For all $n \in \omega$, there exists $f(n) \in \omega$ such that $\|A_{n,\ge f(n)}\|<\infty$;

2. (T3^♯) $\sum \nolimits _kA_{n,k}x_k$ converges in the norm of $Y$ for all $\boldsymbol {x} \in c(X,\,\mathcal {I})$ and $n \in \omega$;

3. (T5^♯) $A \in (c_{00}(X,\,\mathcal {I}),\,c_0(Y,\,\mathcal {J}))$.

It turns out that we obtain a complete characterization if $\mathcal {J}$ is a strongly selective ideal:

Some additional properties can be obtained in special cases: