2. Quasilinear maps and twisted sums in a broader context

Let $X$ and $Y$ be quasi-Banach spaces with quasi-norms $\|\cdot \|_X$ and $\|\cdot \|_Y$. We suppose that $Y$ is a subspace of some vector space $\Sigma$.

Definition 2.1 A map $\Omega :X\to \Sigma$ is called quasilinear from $X$ to $Y$ with ambient space $\Sigma$ and denoted $\Omega : X\curvearrowright Y$ if it is homogeneous and there exists a constant $C$ so that for each $x,z\in X$,

\[ \Omega(x+z)-\Omega x-\Omega z\in Y\quad \textrm{and}\quad \|\Omega(x+z)-\Omega x-\Omega z\|_Y \leq C(\|x\|_X +\|z\|_X\!). \]

The role of the ambient space was considered in [Reference Castillo and Ferenczi10]. There it was shown that given two quasilinear maps $\Phi, \Psi :X\curvearrowright Y$, the former with ambient space $A$ and the second with ambient space $B$ then one can consider that both maps have a certain common space $C$ as ambient space. The main implication of this is that $\Phi + \Psi$ exists.

A quasilinear map $\Omega : X\curvearrowright Y$ generates a quasi-Banach twisted sum of $Y$ and $X$ defined as

\[ Y\oplus_\Omega X =\{(\beta,x)\in \Sigma\times X : \beta-\Omega x\in Y\} \]

and endowed with the quasi-norm $\|(\beta,x)\|_\Omega =\|\beta -\Omega x\|_Y+\|x\|_X$. Indeed,

(2.1)

where $\imath _1 (y) =(y,0)$ and $\pi _2(\beta,x)=x$ is an exact sequence and $\|(y,0)\|_\Omega =\|y\|_Y$ and $\|(\Omega x,x)\|_\Omega =\|x\|_X$.

If $\Omega$ is bounded then $Y\oplus _\Omega X=Y\times X$ and $\|y-\Omega x\|_Y +\|x\|_X$ and $\|y\|_Y+\|x\|_X$ are equivalent quasi-norms on this space. If $\Omega$ is trivial then $Y\oplus _\Omega X$ is isomorphic to $Y\oplus X$.

The map $B_\Omega : X\to Y\oplus _\Omega X$ defined by $B_\Omega x =(\Omega x,x)$ is a bounded homogeneous selection for $\pi _2$ such that $\|B_\Omega x\|_\Omega = \|x\|_X$ for each $x\in X$. If $B_1: X\to Y\oplus _\Omega X$ is another homogeneous bounded selection for $\pi _2$ then it has the form $B_1 x = (\Omega _1 x,x)$ and $\Omega _1:X\curvearrowright Y$ is a quasilinear map boundedly equivalent to $\Omega$: indeed, given $x,z\in X$, since $B_1(x+z)-B_1x-B_1z =(\Omega _1 (x+z)- \Omega _1 x-\Omega _1 z,0)\in Y\oplus _\Omega X$, we get $\Omega _1 (x+z)- \Omega _1 x-\Omega _1 z\in Y$ and

\begin{align*} \|\Omega_1 (x+z)- \Omega_1 x-\Omega_1 z\|_Y & =\|B_1(x+z)-B_1(x)-B_1z\|_\Omega\\ & \leq C'\left(\|B_1(x+z)\|_\Omega +\|B_1(x+z)\|_\Omega\right)\\ & \leq C''(\|x\|_X +\|z\|_X\!). \end{align*}

Moreover, for each $x\in X$, we have $(\Omega x,x), (\Omega _1 x,x)\in Y\oplus _\Omega X$, hence $\Omega x-\Omega _1 x\in Y$ and $\|\Omega x-\Omega _1 x\|_Y= \|(\Omega x-\Omega _1 x,0)\|_\Omega \leq C(\Omega )(1+\|B_1\|)\|x\|_X$.

The notions of domain and range of centralizer-like mappings $\Omega : X\curvearrowright Y$ have been considered in [Reference Castillo, Corrêa, Ferenczi and González7–Reference Castillo and Ferenczi10].

The domain and range of $\Omega$ are independent of the choice of the bounded selection chosen to define $\Omega$. More precisely:

Quite obviously,

If $\imath _2 x=(0,x)$ and $\pi _1(\beta,x)=\beta$ then one has an exact sequence

(2.2)

Note that $\|(0,x)\|_\Omega =\|x\|_D$, $\|\beta \|_R\leq \|(\beta,x)\|_\Omega$ and $\pi _1$ is surjective with $\ker \pi _1=\textrm {Im}\, \imath _2$.

Proof. Since $\{x\in X :(0,x)\in Y\oplus _\Omega X\}$ coincides with $\ker \pi _1$, it is closed in $Y\oplus _\Omega X$. A more pedestrian proof is possible: Suppose that $(0,x_n)$ converges to $(\beta,x)$ in $Y\oplus _\Omega X$. Then $\beta \in \mathrm {Ran}\,\Omega$ and

\[ \lim_{n\to \infty}\|\beta-\Omega(x_n-x)\|_Y + \|x_n-x\|_X =0, \]

hence $\beta -\Omega (x_n-x)\to 0$ in $Y$ and $x_n-x\to 0$ in $X$. By proposition 2.6, $\Omega (x_n-x)\to 0$ in $\mathrm {Ran}\,\Omega$. Since $\|y\|_R\leq \|(y,0)\|_\Omega =\|y\|_Y$ for each $y\in Y$, we get $\beta =0$.

3. Part I: Inversion

We want to introduce the quasilinear map defining the sequence (2.2) and call it $\Omega ^{-1}$. But, to preserve boundedness we need to be more specific:

Proof. It is clear that, by construction, both maps $M$ and $J$ define quasilinear maps $\mathrm {Ran}\, \Omega \curvearrowright \mathrm {Dom}\, \Omega$ with ambient space $X$. Indeed, given $\sigma, \mu \in \mathrm {Ran}\, \Omega$, $M(\sigma +\mu )-M(\sigma ) - M(\mu ) \in \mathrm {Dom}\, \Omega$ if only we prove that $\Omega ( M(\sigma +\mu )-M(\sigma ) - M(\mu ) )\in Y$. Since $(\sigma, M\sigma ), (\mu, M\mu ), (\sigma + \mu, M(\sigma + \mu )$ belong to $Y\oplus _\Omega X$ then also $(0, M\sigma + M\mu - M(\sigma + \mu )$ is in $Y\oplus _\Omega X$, which is what we want. Analogously with $J$.

To check their bounded equivalence, we show first that if $\omega \in \mathrm {Ran}\, \Omega$ then $M \omega - J\omega \in \mathrm {Dom}\, \Omega$. To get that it is enough to check that $(\omega, J\omega )\in Y\oplus _\Omega X$, as it clearly follows from $\|\omega - \Omega J \omega \|\leq K\textrm {dist}(\omega, \mathrm {Dom}\, \Omega )<\infty$, since then $(0, M\omega - J\omega )= (\omega, M\omega ) - (\omega, J\omega ) \in Y\oplus _\Omega X$. Finally

\begin{align*} \|M \omega - J \omega\| & = \|\Omega(M \omega - J \omega)\| \\ & =\|\omega - \Omega(J\omega) - \omega + \Omega(M\omega) - \Omega(M\omega) + \Omega(J\omega) + \Omega(M \omega - J \omega)\| \\ & \leq\|\omega - \Omega(J\omega)\| + \|\omega -\Omega(M\omega)\| + \|M\omega\| + \|J\omega\| \\ & \leq \|(\omega, M\omega)\| + \|\omega -\Omega(J\omega)\| + \|J\omega\| \\ & \leq K\|\omega\| + \|(\omega, J\omega)|\|\\ & \leq K\|\omega\| + K \mathrm{dist}((\omega, J\omega), \mathrm{Dom}\, \Omega)\\ & \leq K\|\omega\| + K \|(\omega, J\omega) - (0, J\omega - M\omega)\\ & \leq K\|\omega\| + K^2\|\omega\|.\end{align*}

□

In [Reference Cabello Sánchez2], the map $\Omega ^{-1}$ was denoted $\mho$. Propositions 3.3 and 3.4 below show that $\Omega ^{-1}$ can be seen as an inverse of $\Omega$. One has:

We can thus form the exact sequence

(3.1)

with $\imath _1x =(x,0)$ and $\pi _1(x,\beta )=\beta$ as usual; and show it is equivalent to (2.2): The map $\Omega ^{-1}$ defines the exact sequence (2.2) and, moreover:

Proposition 3.4 There is a commutative diagram

Proof. It is enough to observe that $U(\beta,x)=(x,\beta )$ is a bijective operator that makes commutative the diagram. Indeed, $U$ is well defined: if $(\beta,x)\in Y\oplus _\Omega X$ then $\beta \in \mathrm {Ran}\,\Omega$ and $(\beta,\Omega ^{-1}\beta )\in Y\oplus _\Omega X$. Thus, $(0, x-\Omega ^{-1}\beta )\in Y\oplus _\Omega X$, hence $x-\Omega ^{-1}\beta \in \mathrm {Dom}\,\Omega$ and $(x,\beta )\in \mathrm {Ran}\,\Omega \oplus _{\Omega ^{-1}} \mathrm {Dom}\,\Omega$.

$U$ is surjective: if $(x,z)\in \mathrm {Ran}\,\Omega \oplus _{\Omega ^{-1}} \mathrm {Dom}\,\Omega$ then $z\in \mathrm {Ran}\,\Omega$ and $x-\Omega ^{-1}z\in \mathrm {Dom}\,\Omega$. Thus, $(z,\Omega ^{-1}z), (0,x-\Omega ^{-1}z)\in Y \oplus _\Omega X$, hence $(z,x)\in Y\oplus _\Omega X$.

$U$ is bounded: $\|U(\beta,x)\|_{\Omega ^{-1}}= \|(x,\beta )\|_{\Omega ^{-1}}= \|(x-\Omega ^{-1}\beta \|_D +\|\beta \|_R$. Since $\|\beta \|_R\leq \|(\beta,x)\|_\Omega$ and

\begin{align*} \|x-\Omega^{{-}1}\beta\|_D & =\|(0,x-\Omega^{{-}1}\beta)\|_\Omega = \|(\beta,x)-(\beta,\Omega^{{-}1}\beta)\|_\Omega\\ & \leq C\|(\beta,x)\|_\Omega+ C\|(\beta,\Omega^{{-}1}\beta)\|_\Omega \leq C\|(\beta,x)\|_\Omega+ C\|B_p\| \|\beta\|_R, \end{align*}

we get $\|U(\beta,x)\|_{\Omega ^{-1}}\leq (C+C\|B_p\|+1)\|(\beta,x)\|_\Omega$, and $U^{-1}$ is bounded by the open mapping theorem.

With the same ideas we get that $\Omega$ and $(\Omega ^{-1})^{-1}$ are boundedly equivalent. The dominion and range of a map can be used to detect its boundedness.

Let us consider the diagram

(3.2)

the unlabelled arrows are the natural inclusions. The diagram is commutative since $\pi _1 \imath _1 x =x$ and $\pi _2\imath _2 x=x$.

4. Part II. Duality

To consider duality issues, we must confine ourselves to deal with Banach spaces since quasi-Banach spaces with trivial dual exist. But since a twisted sum of Banach space can be a quasi-Banach (non-Banach) space, we must also restrict ourselves to consider only exact sequences $0\longrightarrow Y \longrightarrow Z \longrightarrow X \longrightarrow 0$ in which the three terms $Y,Z,X$ are Banach spaces. When $\Omega : X\curvearrowright Y$ is quasilinear, then we only know that $Y\oplus _\Omega X$ is a quasi-Banach space. To guarantee that it is isomorphic to a Banach space, we must impose to the map $\Omega$ an additional condition:

If $\Omega$ is $1$-quasilinear then $\|\cdot \|_\Omega$ is equivalent to a norm, and thus $X\oplus _\Omega Y$ is a Banach space. Kalton showed [Reference Kalton15] that quasilinear maps on $B$-convex Banach spaces are $1$-quasilinear while Kalton and Roberts [Reference Kalton and Roberts18] showed that quasilinear maps on $\mathcal {L}_\infty$ spaces are $1$-quasilinear. We will say that a quasilinear map $\Omega :X\curvearrowright Y$ has dense domain if $\mathrm {Dom}\,\Omega$ is a dense subspace of $X$.

Proof. Let $(x,y)\in \mathrm {Dom}\,\Omega \times Y$. Then

\[ \|J(x,y)\|_\Omega =\|y-\Omega x\|_Y+\|x\|_X\leq C(\|y\|_Y+\|\Omega x\|_Y\!)+\|x\|_X \leq C(\|x\|_D+\|y\|_Y\!). \]

For the second part, let $(\beta,x)\in Y\oplus _\Omega X$ and $\varepsilon >0$, and consider the exact sequence (2.1). Since $\mathrm {Dom}\,\Omega$ is dense in $X$, we can select $z\in \mathrm {Dom}\,\Omega$ such that $\|q(\beta,x)-z\|_X<\varepsilon$. Then $(\beta,x-z)\in Y\oplus _\Omega X$ and $\textrm {dist}((\beta,x-z),\ker q)<\varepsilon$. So we can choose $y\in Y$ such that $\|(\beta,x)-(y,z)\|<\varepsilon$.

Consider the dual sequences of (2.1) and (2.2), namely:

(4.1)

(4.2)

By proposition 4.2, the conjugate operator $J^*:(Y\oplus _\Omega X)^*\to (\mathrm {Dom}\,\Omega )^*\times Y^*$ is continuous and injective. Therefore, if $F\in (Y\oplus _\Omega X)^*$ then $J^*F= (\alpha ^*,y^*)\in (\mathrm {Dom}\,\Omega )^*\times Y^*$.

Proof. Let $F\in (Y\oplus _\Omega X)^*$, $y\in Y$ and $x\in \mathrm {Dom}\,\Omega$. Then

\begin{align*} \langle j^*F,y \rangle & = \langle F,jy \rangle = \langle F,(y,0) \rangle = \langle F,J(0,y)\rangle\\ & = \langle J^*F,(0,y)\rangle = \langle (\alpha ^*,y^*),(0,y)\rangle =\langle y^*,y\rangle, \end{align*}

and similarly $\langle i^*F,x \rangle = \langle \alpha ^*,x\rangle$.

We pass to the definition of the dual $1$-quasilinear map. Since $X^*$ is a subspace of $(\mathrm {Dom}\,\Omega )^*$, Proposition 4.4 allows us to define a map $\Omega ^*$ as follows:

Proof. Since $B$ is a selector for $\imath _1^*$, for every finite set $y_1^*, \ldots, y_n^*$ one has $\sum B(y_i^*)- B(\sum y_i^*) \in \ker \imath _1^*=\textrm {Im}\, \pi _2^*$ and thus

\begin{align*} J^*\sum B(y_i^*)- J^*B\left(\sum y_i^*\right)& = \sum \left( \Omega^*(y_i^*), y_i^*\right) - \left(\Omega^*\left(\sum y_i^*\right), \sum y_i^*\right)\\ & =\left( \sum \Omega^*(y_i^*) - \Omega^*\left(\sum y_i^*\right),0 \right)\\ & \in X^*\oplus_{\Omega^*} Y^* \end{align*}

hence $\sum \Omega ^*(y_i^*) - \Omega ^*(\sum y_i^*)\in X^*$ and $\|\sum \Omega ^*(y_i^* )-\Omega ^*(\sum y_i^*)\|_{X^*}\leq 2\|B\|\sum \|y_i^*\|_{Y^*}.$

We claim that $X^*\oplus _{\Omega ^*} Y^*= J^*((Y\oplus _\Omega X)^*)$. Indeed, if $(\alpha ^*,y^*)\in X^*\oplus _{\Omega ^*} Y^*$ then $\alpha ^*-\Omega ^*y^*\in X^*$. Thus, $(\alpha ^*-\Omega ^*y^*,0)\in J^*((Y\oplus _\Omega X)^*)$, hence $(\alpha ^*,y^*) =(\alpha ^*-\Omega ^*y^*,0) +J^*By^* \in J^*((Y\oplus _\Omega X)^*)$. Conversely, if $(\alpha ^*,y^*)=J^* F$ for some $F\in (Y\oplus _\Omega X)^*$ then $\imath _1^*F=y^*$. Thus, $J^*(F-By^*)=(\alpha ^*-\Omega ^*y^*,0)\in \ker \imath _1^*$, hence $\alpha ^*-\Omega ^*y^*\in X^*$, and we get $(\alpha ^*,y^*)\in X^*\oplus _{\Omega ^*} Y^*$.

Now we consider the formal identity map $W:X^*\oplus _{\Omega ^*} Y^*\to (\mathrm {Dom}\,\Omega )^*\times Y^*$ defined by $W(\alpha ^*,y^*)= (\alpha ^*,y^*)$. Since $W(\alpha ^*,y^*)= (\alpha ^*-\Omega ^*y^*,0) +J^*B y^*$, we get

\begin{align*} \|W(\alpha^*,y^*)\| & \leq \|(\alpha^*-\Omega^*y^*,0)\| +\|J^*By^*\|\\ & \leq \|\alpha^*-\Omega^*y^*\|_{X^*}+\|J^*\|\cdot\|B\|\cdot \|y^*\|_{Y^*}\\ & \leq \|J^*\|\cdot\|B\|\cdot \|(\alpha^*,y^*)\|_{\Omega^*}. \end{align*}

Thus, $W$ is continuous. Since $J^{*-1}$ is a closed operator, $V=J^{*-1}W$ is a closed bijective operator from $X^*\oplus _{\Omega ^*} Y^*$ onto $(Y\oplus _\Omega X)^*$. By the closed graph theorem, $V$ is an isomorphism.

We will refer to $\Omega ^*$ as the adjoint of $\Omega$. We have the following duality relations.

The next result shows that $(\Omega ^*)^{-1}$ and $(\Omega ^{-1})^*$ are equivalent.

Theorem 4.8 The exact sequence

generated by the $1$-quasilinear map $(\Omega ^{-1})^*$ and the exact sequence

generated by $(\Omega ^*)^{-1}$ are equivalent.

Proof. Proposition 4.7 allows us to write the diagram

and theorem 4.6 shows that $J^*$ makes it commutative: $J^*\pi _1^* = (\pi _1 J)^*$ and $\pi _1 J^* = \imath _2^*$.

We describe next the duality action of $X^*\oplus _{\Omega ^*} Y^*$ on (a dense subspace of) $Y\oplus _\Omega X$.

Proof. For the first part, $\langle V(\alpha ^*,y^*),J(z,y)\rangle = \langle (\alpha ^*,y^*),(z,y)\rangle = \langle \alpha ^* ,z \rangle + \langle y^*,y\rangle$. For the second part, note that $(\Omega z,z)\in Y\oplus _\Omega X$, $(\Omega ^* y^*,y^*)\in X^*\oplus _{\Omega ^*} Y^*$ and

\begin{align*} \left|\langle y^*,\Omega z\rangle+ \langle\Omega^* y^*,z\rangle \right| & = \left|\langle V(\Omega^* y^*,y^*),(\Omega z,z)\rangle\right|\\ & \leq \|V\|\cdot\|(\Omega^* y^*,y^*)\|_{\Omega^*}\|(\Omega z,z)\|_\Omega\\ & \leq \|V\|\cdot\|B_{j^*}\|\cdot\|y^*\|_{Y^*}\|z\|_X.\end{align*}

□

Summing up since any exact sequence $0\longrightarrow Y \longrightarrow Z \longrightarrow Z \longrightarrow 0$ of Banach spaces has a dual sequence $0\longrightarrow X^* \longrightarrow Z^* \longrightarrow X^* \longrightarrow 0$, one can associate a classical $1$-quasilinear map $\Omega ^*: Y^*\longrightarrow X^*$ to any classical $1$-quasilinear map $\Omega : X\longrightarrow Y$. The quasilinear map $\Omega ^*$ exists but is not unique. A different thing is what occurs with $1$-quasilinear maps $X\curvearrowright Y$ with ambient space $\Sigma$ as those we are dealing with in this paper. Let us first revisit the so-called Kalton duality explained in [Reference Castillo and Moreno11] for classical quasilinear maps

Definition 4.10 Two $1$-quasilinear maps $\Omega : B \to A$ and $\Phi : A^*\to B^*$ are called bounded duals to one of the other if there is $C=C(\Omega, \Phi )>0$ such that for every $b\in B, a^*\in A^*$ one has

\[ |\langle \Omega b, a^*\rangle + \langle b, \Phi a^*\rangle |\leq C \|b\|\|a^*\|. \]

It is easy to check that if $\Phi, \Psi$ are both bounded duals of $\Omega$ then $\Phi - \Psi$ is bounded:

\begin{align*} \left \langle (\Phi - \Psi)(x), y\rangle\right| & = \left \langle \Phi x, y\rangle + \langle x, \Omega y \rangle - \langle x, \Omega y\rangle - \langle \Psi x, y\rangle\right|\\ & \leq (C(\Phi, \Omega) +C(\Psi, \Omega)) \|x\|\|y\| \end{align*}

Proposition 4.11 If $\Omega$ generates the exact sequence $0\to A \stackrel {\imath }\to A\oplus _\Omega B \stackrel {\pi }\to B \to 0$ and $\Phi$ is a bounded dual of $\Omega$ then $\Phi$ generates the dual sequence $0\to B^*\to (A\oplus _{\Omega } B)^* \to A^* \to 0$, with the meaning that there is an operator $D: B^*\oplus _\Phi A^* \longrightarrow (A\oplus _\Omega B)^*$ given by

\[ \langle D(b^*, a^*), (a,b)\rangle = \langle b^*, b\rangle + \langle a^*, a \rangle \]

making a commutative diagram

In particular, the spaces $B^*\oplus _\Phi A^*$ and $(A\oplus _\Omega B)^*$ are isomorphic via the pairing

\[ \langle (b^*, a^*), (a,b)\rangle = \langle b^*, b\rangle + \langle a^*, a \rangle \]

Proof. The continuity of $D$ is:

\begin{align*} \langle b^*, b\rangle + \langle a^*, a \rangle & = \langle b^*, b\rangle + \langle a^*, a -\Omega b\rangle + \langle a^*, \Omega b\rangle \\ & = \langle b^* - \Phi a^*, b\rangle + \langle \Phi a^*,, b\rangle + \langle a^*, a -\Omega b\rangle + \langle a^*, \Omega b\rangle \\ & \leq C|a^*\| \|b\| + \|a^*\|\|a- \Omega b\| + \|b^*\|\|b^* - \Phi a^*\|\\ & \leq C'\|(a, b)\|_\Omega \|(b^*, a^*)\|_\Phi. \end{align*}

Moreover $D$ makes the diagram commutative: $D\imath = \pi ^*$ since

\[ D\imath(b^*)(a,b) = D(b^*, 0)(a,b) = \langle b^*, b\rangle = b^*\pi(a,b) = \pi^*(b^*)(a,b) \]

while $\imath ^* D = \pi$ since

\[ \imath^* D(b^*, a^*)(a) = D(b^*, a^*)(a,0) = \langle a^*, a\rangle = \pi(b^*, a^*)(a). \]

Finally, $D$ is an isomorphism by the $3$-lemma.

The approach via Kalton duality presents some difficulties in the context of this paper in which quasilinear maps take values in some ambient space. Let us formulate the result in a typical situation: complex interpolation theory. Let $(X_0,X_1)$ be a Banach couple with sum (or ambient space) $\Sigma$ and intersection $\Delta$, which is equipped with the norm $x\in X_0\cap X_1\longmapsto \max \big {(} \|x\|_0, \|x\|_1\big {)}$. We assume that $(X_0,X_1)$ is regular according to Cwikel [Reference Cwikel12], i.e. $\Delta$ is dense in each $X_i$. Cwikel shows in [Reference Cwikel12, Theorem 3.1]:

$\bigstar$ Each $X_i^*$ embeds into $\Delta ^*$ in such a way that $X_0^*\cap X_1^*=\Sigma ^*$.

Let $\Omega : X_0\cap X_1 \longrightarrow \Sigma$ and $\Phi : X_0^*\cap X_1^* \longrightarrow \Sigma$ be homogeneous maps. Observe that the duality conditions $\langle \Omega x, y\rangle$ and $\langle x, \Phi y\rangle$ make sense: $\Omega x\in \Sigma$, $y\in \Sigma ^*$ and $x\in \Delta$, $\Phi y\in \Sigma =\Delta ^*$. Let now $X$ be a Banach space so that $\Delta \subset X\subset \Sigma$ and $\Sigma ^*\subset X^* \subset \Delta ^*$ with $\Delta$ is still dense in $X$ and $\Sigma ^*$ dense in $X^*$. Assume that $\Omega :X \curvearrowright X$ is a quasilinear map with ambient space $\Sigma$ and $\Phi : X^*\curvearrowright X^*$ is a quasilinear map with ambient space $\Delta ^*$. If there exists $C>0$ such that for $x\in X_0\cap X_1=\Delta$, $y\in X_0^* \cap X_1^*=\Sigma ^*$ one has $\left | \langle \Omega x, y\rangle + \langle x, \Phi y\rangle \right | \leq C \|x\|\|y\|$ then $\Omega, \Phi$ are bounded duals to one of each other since there is an isomorphism $D(x^*,y^*)(y,x) = \langle x^*, x\rangle + \langle y^*, y\rangle$ making the diagram

commute. This is a reformulation of what was already proved before and is precisely the context presented in [Reference Cabello Sánchez, Castillo and Corrêa5]. To provide a more general version, we will use the following approach:

Theorem 4.12 Let $X,Y$ be reflexive Banach spaces so that:

• • Both $X$ and $X^*$ embed into an ambient space $\Sigma$ in such a way that there is a subspace $\Sigma _0$ that is dense both in $X$ and $X^*$ and $\Sigma \subset \Sigma _0^*$.

• • Both $Y$ and $Y^*$ embed into an ambient space $\Theta$ in such a way that there is a subspace $\Theta _0$ that is dense both in $Y$ and $Y^*$ and $\Theta \subset \Theta _0^*$.

Let $\Omega : X \curvearrowright Y$ be a $1$-quasilinear map with ambient space $\Theta$ and such that $\Omega [\Sigma _0]\subset \Theta _0$ and let $\Phi : Y^*\curvearrowright X^*$ be a $1$-quasilinear map with ambient space $\Sigma$ and such that $\Phi [\Theta _0]\subset \Sigma _0$. If the following condition holds:

$\bigstar \bigstar$ There is $C>0$ so that for every $x\in \sigma$, $y\in \theta$ one has $\left | \langle \Omega x, y\rangle + \langle x, \Phi y\rangle \right |\leq C \|x\|_X\|y\|_{Y^*}$

then $\Omega$ and $\Phi$ are bounded duals, so that the sequences $0\longrightarrow Y \longrightarrow Y\oplus _\Omega X \longrightarrow X\longrightarrow 0$ and $0\longrightarrow X^* \longrightarrow X^*\oplus _\Phi Y^* \longrightarrow Y^* \longrightarrow 0$ are dual one of the other and the spaces $X^*\oplus _\Phi Y^*$ and $(Y\oplus _\Omega X)^*$ are isomorphic under the duality

\[ \langle (\eta, y^*), (\omega, x) \rangle = \langle \eta, x \rangle + \langle \omega, y^*\rangle. \]

We connect now inversion and duality.

Proof. Since $\Omega$ and $\Phi$ are bounded duals, there is no loss of generality replacing $\Phi$ by $\Omega ^*$. The proof is a drawing, in fact, two:

containing the two representations for $Z$ and their duals for $Z^*$. The exactness of the sequences for $Z$ means $\mathrm {Ran}\, \Omega = Z/\mathrm {Dom}\, \Omega$, hence $(\mathrm {Ran}\, \Omega )^*= (\mathrm {Dom}\, \Omega )^\perp$. Analogously $(\mathrm {Ran}\, \Omega ^{-1})^*= (\mathrm {Dom}\, \Omega ^{-1})^\perp$ and the right diagram becomes

Since the horizontal sequence is the dual of the original sequence $\Omega$ and the vertical is the dual of $\Omega ^{-1}$, the diagram is

Now apply proposition 4.13: use $(\mathrm {Dom}\, \Omega )^\perp = \mathrm {Dom}\, \Omega ^*$ to get $\mathrm {Dom}\, (\Omega ^*)^{-1} = (\mathrm {Dom}\, \Omega ^{-1})^*$ and $(\mathrm {Dom}\, \Omega )^* = \mathrm {Ran}\, \Omega ^*$ to get $\mathrm {Ran}\,(\Omega ^*)^{-1}= (\mathrm {Dom}\, \Omega ^{-1})^* = \mathrm {Ran}\, (\Omega ^{-1})^*$. The horizontal sequence $\Omega ^*$ is now

therefore $\Omega ^* = ((\Omega ^{-1})^{*})^{-1}$ and thus $(\Omega ^*)^{-1} = (\Omega ^{-1})^{*}$.

5. Applications

Given a suitable pair of Banach spaces $(X_0, X_1)$ with ambient space $\Sigma$, one can consider the sequence of Schechter interpolators $\Delta _{k, \theta }(f) =f^{(k)}(\theta )(1/2)/k!$ defined on the suitable Calderón space $\mathcal {C}$ corresponding to the pair, see [Reference Bergh and Löfström1] or [Reference Schechter21] for details. With those ingredients, one can generate, following [Reference Rochberg20], the family of associated Rochberg spaces $\mathcal {R}_n(\theta ) =\{ (\Delta _{n-1, \theta }(f), \ldots, \Delta _{0, \theta }(f)): f\in \mathcal {C}\}$, that generalize the complex interpolation space $\mathcal {R}_1(\theta )=(X_0,X_1)_\theta$ and twisted sum space $\mathcal {R}_2(\theta )= \mathcal {R}_1(\theta )\oplus _{\Omega _\theta } \mathcal {R}_1(\theta )$, where $\Omega _\theta$ is the differential quasilinear map associated to the couple $(\Delta _1, \Delta _0)$. It was shown in [Reference Cabello Sánchez, Castillo and Kalton6] that for each pair $m,k\in \mathbb {N}$ with $n=m+k$, there are natural exact sequences

that can be arranged forming commutative diagrams of exact sequences (we omit from now on the initial and final arrows $0\to \cdot$ and $\cdot \to 0$ and the parameter $\theta$)

Let $\Omega _{m,n}: \mathcal {R}_m \curvearrowright \Sigma ^n$ be the quasilinear map associated with the sequence $0\to \mathcal {R}_n\to \mathcal {R}_{n+m}\to \mathcal {R}_m \to 0$.

We shall consider three situations: first, the case of complex interpolation for the pair $(\ell _\infty, \ell _1)$ in which the involved differentials are not linear or bounded; second, translation operators which are bounded and homogeneous; third, the case of weighted Köthe spaces, in which the differentials are linear unbounded operators.

5.1 Higher-order Kalton–Peck maps

Our results will complete those in [Reference Castillo, Corrêa, Ferenczi and González9]. The interpolators $\delta _{\theta }'$ and $\delta _{\theta }$ that define complex interpolation yield as interpolation space $X_\theta =(\ell _\infty, \ell _1)_\theta =\ell _p$ for $p=\theta ^{-1}$ with associated differential ${\sf K}\hspace {-1pt}{\sf P}: X_\theta \curvearrowright X_\theta$, usually called the Kalton–Peck map,

\[ {\sf K}\hspace{-1pt}{\sf P} (x)= p\; x\log \frac{|x|}{\|x\|_p} \]

and associated twisted sum space $X_\theta \oplus _{{\sf K}\hspace {-1pt}{\sf P}} X_\theta = Z_p$, usually called the Kalton–Peck space [Reference Kalton and Peck17]. According to [Reference Kalton and Peck17, Lemma 5.3 (c)], see also [Reference Castillo, Corrêa, Ferenczi and González9], $\mathrm {Dom}\, {\sf K}\hspace {-1pt}{\sf P}$ is the Orlicz sequence space $\ell _{f_p}$ generated by $f_p(t)=t^p |\log t|^p$, while $\mathrm {Ran}\, {\sf K}\hspace {-1pt}{\sf P}$ can be obtained by duality as the Orlicz space $\ell _{f_p}^* = \ell _{f_p^*}$ generated by the Orlicz conjugate function $f_p^*$ of $f_p$ which, according to [Reference Lindenstrauss and Tzafriri19, Ex. 4.c.1], is equivalent to $g_q(t)=t^q|\log t |^{p(1-q)} = t^q|\log t |^{-q}$ at $0$, for $pq=p+q$. We now follow [Reference Cabello Sánchez, Castillo and Corrêa5, Reference Cabello Sánchez, Castillo and Kalton6, Reference Castillo, Corrêa, Ferenczi and González9] to construct the associated Rochberg spaces. Let us focus at $\theta =1/2$: in this case, each space $\mathcal {R}_n$ is isomorphic to its dual and this allows us to compare the associated differential with its dual and uncover remarkable symmetries. Precisely, $\mathcal {R}_1=\ell _2$, while $\mathcal {R}_2 = Z_2$, the Kalton–Peck space. Let ${\sf K}\hspace {-1pt}{\sf P}_{k,m}: \mathcal {R}_k\curvearrowright \mathcal {R}_m$ the $1$-quasilinear map with ambient space $\ell _\infty ^m$ that generates the sequence $0\longrightarrow \mathcal {R}_m \longrightarrow \mathcal {R}_n \longrightarrow \mathcal {R}_k \longrightarrow 0$. We are not interested here in the properties of the spaces $\mathcal {R}_n$, a topic studied in [Reference Cabello Sánchez, Castillo and Corrêa5, Reference Cabello Sánchez, Castillo and Kalton6, Reference Castillo, Corrêa, Ferenczi and González9], but in the properties of the maps ${\sf K}\hspace {-1pt}{\sf P}_{k,m}$. The case $m=k=1$ is in [Reference Kalton and Peck17]. In general, one has:

Theorem 5.1 The $1$-quasilinear maps ${\sf K}\hspace {-1pt}{\sf P}_{k,m}$ and ${({\sf K}\hspace {-1pt}{\sf P}_{m,k})}^*$ are isomorphic.

Proof. It was proved in [Reference Cabello Sánchez, Castillo and Corrêa5] that for each $n\in \mathbb {N}$ there is an isomorphism $u_n =\mathcal {R}_n\to \mathcal {R}_n^*$ given by

\[ u_n\left(a_{n-i}\right)_{i=1}^{n} = \left(({-}1)^{n-i} a_{n-i}\right)_{i=1}^{n}. \]

such that for each $m,k\in \mathbb {N}$ and $n=m+k$ the following diagram is commutative:

(5.1)

which implies that ${{\sf K}\hspace {-1pt}{\sf P}_{m,k}}^*$ is equivalent to $(-1)^ku_m {\sf K}\hspace {-1pt}{\sf P}_{k,m} u_k^{-1}$.

This yields the estimate:

\begin{align*} & |\langle {\sf K}\hspace{-1pt}{\sf P}_{m,k}(a_0, \ldots, a_{m-1}), (x_0, \ldots, x_{k-1}) \rangle\\ & \qquad + \langle ({-}1)^ku_m {\sf K}\hspace{-1pt}{\sf P}_{k,m} u_k^{{-}1}(x_0, \ldots, x_{k-1}), (a_0, \ldots, a_{m-1})\rangle|\\ & \quad\leq C \|(a_0, \ldots, a_{m-1})\|_{\mathcal{R}_m} \|(x_0, \ldots, x_{k-1})\|_{\mathcal{R}_k} \end{align*}

We know from [Reference Cabello Sánchez, Castillo and Kalton6] that

\begin{align*} {\sf K}\hspace{-1pt}{\sf P}_{1,2}(x) & = 2x(\log^2 \frac{|x|}{\|x\|}, \log \frac{|x|}{\|x\|})\\ {\sf K}\hspace{-1pt}{\sf P}_{2,1}(y,x) & = 2\left(y - 2x\log\frac{|x|}{\|x\|}\right) \log \frac{|y - 2x\log\frac{|x|}{\|x\|}|}{\|y - 2x\log\frac{|x|}{\|x\|}\|} + 2x\log^2\frac{|x|}{\|x\|} \end{align*}

and therefore the estimate $\left | \langle {\sf K}\hspace {-1pt}{\sf P}_{1,2}(x), z\rangle - \langle x, u_1{\sf K}\hspace {-1pt}{\sf P}_{2,1}u_2(z) \rangle \right | \leq C \|x\|_{\ell _2}\|z\|_{Z_2}$ becomes

\begin{align*} & \left |2xz_1\log^2 \frac{|x|}{\|x\|}+ 2xz_2 \log \frac{|x|}{\|x\|}\vphantom{\frac{|z_1 + 2z_2\log\frac{|z_2|}{\|z_2\|}|}{\|z_1 + 2z_2\log\frac{|z_2|}{\|z_2\|}\|}}\right.\\ & \left.\qquad -\, x \left(z_1 - 2z_2\log\frac{|z_2|}{\|z_2\|}\right) \log \frac{|z_1 + 2z_2\log\frac{|z_2|}{\|z_2\|}|}{\|z_1 + 2z_2\log\frac{|z_2|}{\|z_2\|}\|} - 2z_2\log^2\frac{|z_2|}{\|z_2\|} \right|\\ & \quad \leq C \|x\|_{\ell_2}\|(z_1, z_2)\|_{Z_2} \end{align*}

Beware that the $u_n's$ cannot be deleted. For instance, $({\sf K}\hspace {-1pt}{\sf P} x, x)\in Z_2$ while $({\sf K}\hspace {-1pt}{\sf P} x, -x)\notin Z_2$, no matter if $Z_2$ and $u_2[Z_2]$ are isometric.

5.3 Weighted Köthe spaces

Fix a Köthe function space $X$ with the Radon–Nikodym property, let $w_0$ and $w_1$ be weight functions, and consider the interpolation couple $(X_0, X_1)$, where $X_j= X(w_j)$, $j =0,1$ with their natural norms. In [Reference Castillo, Corrêa, Ferenczi and González7, Proposition 4.1], it was shown that $X_\theta = X(w_\theta )$ for $0<\theta <1$, where $w_{\theta } = w_0^{1 - \theta }w_1^{\theta }$. In [Reference Castillo, Corrêa, Ferenczi and González8], it was obtained $X(\omega _0) \cap X(\omega _1) = X(\omega _\vee )$ with $\omega _\vee = \max \{\omega _0, \omega _1\}$; $X(\omega _0) + X(\omega _1) = X(\omega _\wedge )$ with $\omega _\wedge = \min \{\omega _0, \omega _1\}$; $\mathrm {Dom}\,(\Omega _{\Psi, \Phi }) = X(w_\theta ) \cap X(w_\theta \left |\log {w_1}/{w_0}\right |)$ and $\mathrm {Ran}\,(\Omega _{\Psi, \Phi })= X(w_\theta )+ X(w_\theta \left |\log {w_1}/{w_0}\right |^{-1})$. For $\Psi = \Delta '_{\theta }$ and $\Phi = \Delta _{\theta }$, we obtain $\Omega _{\Psi, \Phi }f = \log {w_1}/{w_0}\cdot f$. Consequently, $\Omega _{\Psi, \Phi }^{-1}(x) = \Omega _{\Phi, \Psi }(x) = (\log {\omega _1}/{\omega _0})^{-1} x$. These maps are linear but unbounded, and in this case $\mathrm {Dom}\, \Omega _{\Psi, \Phi }f = \{f\in X(w): \log {w_1}/{w_0}\cdot f \in X(w)\} = X(w\log {w_1}/{w_0})$. Let us consider a specially interesting case already considered in [Reference Castillo, Corrêa, Ferenczi and González9]. Let $w$ be a weight sequence (we will understand as in [Reference Lindenstrauss and Tzafriri19, 4.e.1] a non-increasing sequence of positive numbers such that $\lim w_n=0$ and $\sum w_n=\infty$). We set $w_0=w^{-1}$ and $w_1=w$ and let us consider the interpolation pair $(\ell _2(w^{-1}), \ell _2(w))$, whose complex interpolation space at $1/2$ is $\ell _2$. A homogeneous bounded selector for $\Delta _0$ is $B(x)(z) = w^{2z-1}x$. Thus, $B(x)^{(n}(z) = 2^nw^{2z-1}\log ^n w \cdot x$ and therefore $B(x)^{(n}(1/2) = 2^n\log ^n w \cdot x$. Consequently,

\[ \Omega_{\langle n-1, \ldots, 1\rangle,0}(x) = \left(\frac{2^{n-1}}{(n-1)!}\log^{n-1} w \cdot x, \ldots, 2\log w \cdot x \right) \]

which yields

\[ \mathcal{R}_n = \{(y_{n-1}, \ldots, y_1, x)\in \ell_\infty^{n-1}\times \ell_2: (y_{n-1}, \ldots, y_1) - \Omega_{\langle n-1, \ldots, 1\rangle,0}(x)\in \mathcal{R}_{n-1}\} \]

In particular, $\mathcal {R}_2 = \{(y,x)\in \ell _\infty \times \ell _2: x\in \ell _2, \quad y - 2\log w \cdot x\in \ell _2\}$ is generated by the differential $\Omega _{1,0}(x) = 2\log w \cdot x$. Consequently, $\mathrm {Dom}\, \Omega _{1,0} = \ell _2(\log w)$ and $\mathrm {Ran}\, \Omega _{1,0}=\ell _2(\log ^{-1} w)$ and $\Omega _{1,0}^{-1}(x) = (2\log w)^{-1} \cdot x$. Analogously, it is now straightforward to check that

\[ \mathrm{Dom}\, \Omega_{\langle n-1, \ldots, 1\rangle,0} = \ell_2(\log^{n-1}w). \]