Concept of s-Numbers in Quaternionic Analysis and Schatten Classes

Abstract

In this paper we introduce an axiomatic approach to the theory of s-numbers in quaternionic analysis. To this end, Pietsch’s approach to s-number theory is adapted to the quaternionic framework, following the works of Colombo and Sabadini on quaternionic spectral analysis. One of the main results of this paper is the uniqueness of s-numbers over quaternionic Hilbert spaces. Moreover, examples are given in the quaternionic framework together with the introduction of nuclear numbers. A consequence of the presented theory is a basis independent definition of the Schatten classes over quaternionic Hilbert and Banach spaces.

1 Introduction

The interest in quaternionic analysis can be attributed to the works of Birkhoff and von Neumann. Specifically, in their paper on the logic of quantum mechanics [3], they demonstrated that the Schrödinger equation can be formulated in both complex and quaternionic settings, giving rise to the theory of quaternionic quantum mechanics (QQM). As important examples of applications in the complex setting we remark the study of uncertainty relations or restriction inequalities [10]. Furthermore, these techniques have also application on the study of uniform and pointwise convergence of certain classical transforms [8, 9].

Inspired by this, there has been a tendency to generalize classical theories of analysis to the quaternionic context. In this regard, our objective is to extend the concept of s-numbers to the algebra of the quaternions.

The first instance of s-numbers probably originated from the works of Schmidt [17], where the concept of singular values of integral operators acting on Hilbert function spaces was introduced. Afterwards, in [15, 16], von Neumann and Schatten extended this concept to the setting of compact operators acting on general Hilbert spaces. Hereby, the n-th s-number, was defined to be the n-th eigenvalue of $|S|=\sqrt{S^*S}$ (ordered by algebraic multiplicity), which we denote by $\lambda _n(S)$. These led to the introduction of Schatten classes

$$\begin{aligned} {\mathfrak {S}}_p(H)=\left\{ T\in K(H): \left\| \lambda _n(T)|\ell _p \right\| <\infty \right\} . \end{aligned}$$

Later, an axiomatic approach to the theory of s-numbers was introduced by Pietsch in [20], enabling the extension of s-numbers to general Banach spaces. We utilize this axiomatization to further expand the theory into the quaternionic framework. Unlike in the Hilbert space setting, the s-numbers are not unique in a Banach space, as we will see.

It is currently well established that s-number functions are unique over complex Hilbert spaces. By employing the proposed axiomatizations along with the novel works on quaternionic spectral theory [5], we are able to extend the uniqueness of s-numbers to the quaternionic setting.

Following the same reasoning for the construction of Schatten classes, this approach allowed to extend the results of Neumann/Schatten to a general Banach space. Consequently, various classes of compact operators were obtained, each of which classify the behaviour of certain compact operators with respect to an s-number function. By following this methodology, we derive the quaternionic counterpart of Schatten classes for compact operators acting between quaternionic Banach spaces, thereby establishing a classification for quaternionic compact operators.

Hence, the objective of this work is twofold. Firstly, we aim to extend Pietsch’s axiomatic approach to s-number theory to the quaternionic framework and explore its implications. This is done in the first two sections. Secondly, in the last section, we introduce a suitable definition of Schatten classes on the quaternionic setting.

Some work has already been developed in the direction of suitably defining Schatten classes of quaternionic operators, as seen in [4]. However, the proposed definition is dependent on the choice of a slice on the quaternionic space. This phenomenon is not desired as it implies a dependence on the choice of basis. The proposed axiomatic approach circumvents this issue.

2 Preliminaries

2.1 The Algebra of the Quaternions

The quaternions, denoted by ${\mathbb {H}}$, are the real algebra generated by $\{1,i,j,k\}$ satisfying the following properties:

$$\begin{aligned} i^2 = j^2 = k^2 = -1, \quad ij = -ji = k, \quad jk = -kj = i, \quad ki = -ik = j. \end{aligned}$$

Thus, an element $q \in {\mathbb {H}}$ can be written as $q = x_0 + x_1i + x_2j + x_3k$, where $x_0, x_1, x_2, x_3$ are real values and i, j, k satisfy the above relations. The norm of a quaternion $q \in {\mathbb {H}}$ is defined by:

$$\begin{aligned} \left\| q \right\| =\sqrt{x_0^2+x_1^2+x_2^2+x_3^2}\,. \end{aligned}$$

Finally, the sphere of purely imaginary quaternions is denoted by ${\mathbb {S}}$. More precisely,

$$\begin{aligned} \{q=q_1i+q_2j+q_3k\,:\,q_1^2+q_2^2+q_3^2=1\}. \end{aligned}$$

The noncommutative nature of quaternions implies that a vector space can be considered in a restricted sense. Indeed, one can consider a right vector space¹ which is essentially a vector space where the action of the scalars is only considered from the right. In turn, this allows to define the so called right Banach space.

When dealing with operators, unless otherwise stated, they shall be right linear operators, meaning that for all $s \in {\mathbb {H}}$ and $u, v \in X$ (a right Banach space):

$$\begin{aligned} T(u + v) = T(u) + T(v), \qquad T(us) = T(u)s. \end{aligned}$$

Left linear operators and two sided linear operators are defined in a similar manner, over left modules and two-sided modules, respectively. It is worth noting that whether T is a right and left linear operator there holds $aT = Ta$ for $a\in {\mathbb {R}}$.

To discuss the boundedness of operators, we introduce some additional notation. Let X be a right ${\mathbb {H}}$-Banach space with norm $\left\| \cdot \right\|$. We denote $B^R(X)$ as the two-sided vector space of all right linear bounded operators on X, and $B^L(X)$ as the two-sided vector space of all left linear bounded operators on V. Finally, B(X) will denote the two-sided vector space of two-sided linear operators on X.

In analogy with the classic theory, it can be shown that $B^R(X)$ and $B^L(X)$ are Banach spaces when equipped with their natural norms:

$$\begin{aligned} \left\| T \right\| := \sup _{x\in X}\frac{\left\| T(x) \right\| }{\left\| x \right\| }. \end{aligned}$$

For a given Banach Space X, its unit ball shall be denoted by $B_X$. Moreover, unless otherwise stated, all spaces shall be considered right ${\mathbb {H}}$-Banach spaces.

Remark

We underline that dual structures, subsets of either $B^R$ or $B^L$, have a slight difference in comparison with the classic case of real (complex) spaces. Indeed, considering a right Banach space, its dual space, denoted as $X'$, is the quaternionic left Banach space consisting of all bounded right linear mappings from X to ${\mathbb {H}}$. Consequently, its bidual will be the quaternionic right Banach space of all bounded left linear mappings from $X'$ to ${\mathbb {H}}$. The underscript notation shall be omitted whenever the side of the action of ${\mathbb {H}}$ is clear. The next natural step is to define the dual operator $T'$ of an operator T.

Definition 2.1

Let X and Y be quaternionic right Banach spaces, and let $T\in B^R(X, Y)$. The dual operator of T, denoted as $T'$, is a left linear operator that maps $Y'$ to $X'$. Its action is defined as follows: for $\varphi \in Y'$, if $T'(\varphi )=\psi$, then $\psi (x)=\varphi (T(x))$.

Remark

Observe that, according to this definition, it follows that $T'\in B^L(Y',X')$. Moreover, in the case of Hilbert spaces, the dual operator coincides with is the adjoint operator, denoted by $T^*$.

2.2 Sequence Spaces

Let X be a right ${\mathbb {H}}$-Banach space and consider a sequence $x=(x_n)_{n\in {\mathbb {N}}}$, for which $x_n\in X$, for each $n\in {\mathbb {N}}$.

The sequence x is said to be absolutely p-summable, for $1\le p<\infty$, whenever

$$\begin{aligned} \left\| x|\ell _p \right\| :=\left( \sum _{n=1}^\infty \left\| x_n \right\| _X^p\right) ^\frac{1}{p}<\infty . \end{aligned}$$

If $p=\infty$ then one requires $\left\| x|\ell _\infty \right\| :=\sup _{n\in {\mathbb {N}}}|x_n| <\infty .$ The set of these sequences is denoted by $\ell _p$. Moreover, the sequence x is said to be weakly p-summable if

$$\begin{aligned} (\langle \varphi ,x_n\rangle )_{n\in {\mathbb {N}}}\in \ell _p,\quad \text {for every } \varphi \in X'. \end{aligned}$$

We will mostly be dealing with sequences that are weakly p-summable on the dual unit ball, therefore we will use the following notation

$$\begin{aligned} \left\| x|w_p(X) \right\| :=\sup \left\{ \left\| (\left\langle \varphi ,x_n \right\rangle )_{n\in {\mathbb {N}}}|\ell _p \right\| : \varphi \in B_{X'} \right\} . \end{aligned}$$

2.3 Quaternionic Spectral Theory and Quaternionic Functional Calculus

The results presented in this section are taken from [5] and [6] to which we refer for the corresponding proofs. We start with two useful decompositions of operators.

Theorem 2.2

(Polar representation) [5, pp. 199–201] Every operator $T\in B^R(H)$ admits a unique factorization

$$\begin{aligned} T=UP \end{aligned}$$

(2.1)

into the product of a positive operator P and a partial isometry U on ran(P). The operator P is furthermore given by $P=(T^*T)^\frac{1}{2}$, and ran$(U)=$ran(T).

The first instance where the construction of spectral measures deviates from the classical construction is a consequence of the following Lemma.

Lemma 2.3

(Teichmüller decomposition) [5, p. 201] Let $T\in B^R(H)$. Then there exists a triple (A, J, B) of mutually commuting operators in $B^R(H)$ all of which commute with T such that

$$\begin{aligned} T=A+JB \end{aligned}$$

(2.2)

where, A is self adjoint, B is a positive operator and J is an antiselfadjoint partial isometry operator which is a partial isometry on ker$(T-T^*)^\perp$. The operators A and B are given by

$$\begin{aligned} A=\frac{1}{2}(T+T^*),\quad B=\frac{1}{2}|T-T^*| \end{aligned}$$

(2.3)

and J is the partial symmetry that appears in the polar decomposition of the operator $\frac{1}{2}|T-T^*|$. Finally, the adjoint of T, $T^*$ satisfies $T^*=A-JB$, and every operator in $B^R(H)$ commutes with T and $T^*$ if and only if it commutes with A, B and J.

The second instance, where the quaternionic spectral theory deviates from the classical theory is the notion of spectrum. As noted in [11] the classic notion of spectrum is ill-defined. Only in 2006, the appropriate definition of the spectrum of a linear and bounded operator in the quaternionic setting, known as the S-spectrum was found. The S-spectrum [5, p. 57] provides a suitable extension of the classical spectrum to the quaternionic framework and will be utilized throughout this presentation. More precisely, the S-spectrum is defined as follows:

$$\begin{aligned} \sigma _S(T)=\{s\in {\mathbb {H}}: \underbrace{T^2-2Re(s)T+|s|^2Id}_{:={\mathcal {Q}}_s(T)} \text { is not invertible}\}. \end{aligned}$$

In accordance with the classic theory, the following result shows that some desired properties of the classic notion of spectrum are still preserved when considering the S-spectrum.

Lemma 2.4

[5, p. 193] Let T be a right linear, self-adjoint and bounded operator acting between right ${\mathbb {H}}$-Hilbert spaces. Then $\sigma _S(T)\subseteq {\mathbb {R}}$. Additionally, if T is a positive operator, $\sigma _S(T)\subseteq [0,\infty )$.

2.3.1 S-Functional Calculus

For an exposition on the construction of the spectral measure associated to a quaternionic operator, we refer to [5, p. 234–241]. Let T be a normal operator in $B^R(H)$ and let j be a fixed imaginary unit. Here and thereafter, let $\Omega _j^+ = \sigma _S(T) \cap {\mathbb {C}}_j^+$. Here, ${\mathbb {C}}_j^+$ denotes de complex upper half-plane, i.e. ${\mathbb {C}}_j^+=\{z_0+jz_1\in {\mathbb {C}}_j|z_1\ge 0\}$, where ${\mathbb {C}}_j=\{u+jv\,| u,v\in {\mathbb {R}}\}$ for a given $j\in {\mathbb {S}}$

It follows that, for each $g\in C(\Omega _j^+,{\mathbb {R}})$ and for every $x,y\in H$, there is a unique ${\mathbb {H}}$-valued measure $\mu _{x,y}$ for which

$$\begin{aligned} \left\langle g(T)x,y \right\rangle =\int _{\Omega _j^+} g(p)\,d\mu _{x,y}(p). \end{aligned}$$

Then, one constructs what is called a spectral measure, which is an operator $E(\theta )\in B^R(H)$ such that, for each Borel set $\theta \in \Omega _j^+$, there holds

$$\begin{aligned} \mu _{x,y}(\theta ) = \langle x,E(\theta )y \rangle . \end{aligned}$$

(2.4)

Thus, for a normal operator $T\in B^R(H)$, if $g\in C(\Omega _j^+,{\mathbb {R}})$ one has

$$\begin{aligned} \left\langle g(T)x,y \right\rangle =\int _{\Omega _j^+}g(p) \,d\left\langle E(p)x,y \right\rangle ,\qquad \forall x,y\in H. \end{aligned}$$

(2.5)

This can be further extended. Let us denote by $SC(\Omega )$ the set of continuous intrinsic slice functions. These are the continuous functions $f\in \Omega \rightarrow {\mathbb {H}}$ for which

$$\begin{aligned} \forall q=u+jv\in \Omega :\quad f(q)=f_0(u,v)+jf_1(u,v), \end{aligned}$$

where the functions $f_0,\,f_1$ must satisfy

$$\begin{aligned} f_0(u,v)=f_0(u,-v),\quad f_1(u,-v)=-f_1(u,v) \end{aligned}$$

and be real valued and the set U must be axially symmetric, i.e., for every $q\in \Omega$,

$$\begin{aligned} \{q_0+j|Im (q)|:\, j\in {\mathbb {S}}\}\subset \Omega . \end{aligned}$$

Finally, $SC_j(\Omega _j^+):=\left\{ f\big |_{\Omega _j^+},\, f\in SC(\Omega )\right\}$.

Moreover, if $f=f_0+jf_1\in SC_j(\Omega _j^+)\simeq SC(\Omega )$ then, with the help of Lemma 2.3, for all $x,y\in H$,

$$\begin{aligned} \left\langle f(T)x,y \right\rangle =\int _{\Omega _j^+}f_0(p) \,d\left\langle E(p)x,y \right\rangle +\int _{\Omega _j^+}f_1(p) \,d\left\langle JE(p)x,y \right\rangle . \end{aligned}$$

(2.6)

Here, in both cases, E is the spectral measure on ${\mathfrak {B}}(\Omega _j^+)$ defined above. We will use the simplified notations

$$\begin{aligned} g(T)=\int _{\Omega _j^+}g(p) \,E(dp) \text { and } f(T)=\int _{\Omega _j^+}f_0(p) \,E(dp)+\int _{\Omega _j^+}f_1(p) \,JE(dp) \end{aligned}$$

when referring to (2.5) and (2.6), respectively. In particular, for a normal $T\in B^R(H)$ we have the representation

$$\begin{aligned} T=\int _{\Omega _j^+}Re(p)\,E(dp)+\int _{\Omega _j^+}Im(p) \,JE(dp) \end{aligned}$$

(2.7)

Remark

We can notice the consistency of this construction with the classical theory by observing how (2.6) collapses to the classical case. As we have already seen, even in the quaternionic setting, the spectrum of a positive operator is positive, i.e., $\Omega _j^+=\sigma (T)\subseteq [0,\infty )$. Additionally, the Teichmüller decomposition of a positive operator $T\in B^R(H)$ implies that $J\equiv 0$. This means that, in the case of a positive operator, (2.6) can be written as:

$$\begin{aligned} T=\int _{\sigma (T)} p\,E(dp). \end{aligned}$$

(2.8)

Therefore, for positive operators, the results from the classical theory can be directly applied.

2.4 Theory of Banach Spaces

A great survey on the theory of Banach spaces that will be useful in this section is [1].

In what follows, let X and Y denote general sets. Moreover, let $Y_0$ be a subset of Y and $X_0$ be a closed subset of X. We denote the natural injection from $X_0$ to X and the canonical surjection from Y to the quotient space $Y/Y_0$ respectively as follows:

The following definitions and corresponding results, which we directly adapt to the quaternionic setting from [21, pp. 26–28]. Assume that X and Y are right ${\mathbb {H}}$-Banach spaces and take $T \in B^R(X,Y)$. We define the injection modulus as

$$\begin{aligned} j(T):=\sup \{\tau \ge 0: \left\| Tx \right\| \ge \tau \left\| x \right\| ,\,\forall x\in X\}, \quad \text {and}\quad j(0)=0. \end{aligned}$$

An operator is called an injection if $j(T)>0$. Moreover, if $\left\| T \right\| =j(T)=1$, then T is said to be a metric injection. Analogously we define the surjection modulus

$$\begin{aligned} q(T):=\sup \{\tau \ge 0:T(B_X)\supseteq \tau B_Y\}, \quad \text {and}\quad q(0)=0. \end{aligned}$$

An operator is called a surjection if $q(T)>0$. If $\left\| T \right\| =q(T)=1$ we say that T is a metric surjection. Note that in both cases of metric injection and metric surjection, we require the operator to be an isometry.

The following result is of significant importance for the sequel. Its corollary will play a major role in establishing duality relations between different s-numbers. For completeness, the proof is presented following the exact same lines as the ones presented in [21, p. 26].

Theorem 2.5

Let $T\in B^R(X,Y)$. Then $q(T')=j(T)$ and $j(T')=q(T)$.

Proof

The proof is divided two steps. In the first step, we demonstrate that the surjection modulus does not depend on whether T is a closed operator or not. We utilize this in the second step to establish the desired equalities.

Step 1: Define $\overline{q}(T)=\sup \{\tau \ge 0: \overline{T(B_X)}\supseteq \tau B_Y\}$. It is evident that $q(T)\le \overline{q}(T)$. Without loss of generality, let us assume $\overline{q}(T)>0$. Let $y\in B_Y$, $0<\epsilon <1$, and set $\tau =(1-\epsilon )\overline{q}(T)$ and $y_1=y$. Now, we construct a family of elements in X that will establish the remaining inequality. Inductively, choose $x_1,x_2,\dots \in X$ such that

$$\begin{aligned} \left\| y_k-\tau ^{-1}Tx_k \right\| \le \epsilon ^k\text { and }\left\| x_k \right\| \le \left\| y_k \right\| , \end{aligned}$$

where $y_k=y-\sum _{i=1}^{k-1}\tau ^{-1}Tx_i$. It is clear that $\left\| x_1 \right\| \le \left\| y_1 \right\| \le 1$, and

$$\begin{aligned} \left\| x_k \right\| \le \left\| y_k \right\| =\left\| y_{k-1}-\tau ^{-1}Tx_{k-1} \right\| \le \epsilon ^{k-1}\text { for } k\ge 2. \end{aligned}$$

Therefore, $\sum _{k=1}^{\infty } \left\| x_k \right\| \le \frac{1}{1-\epsilon }$. Let $x=\sum _{k=1}^{\infty } x_k$. It follows that

$$\begin{aligned} y=\sum _{i=1}^{\infty } \tau ^{-1}Tx_k=\tau ^{-1}Tx, \end{aligned}$$

and since $\left\| x \right\| \le (1-\epsilon )^{-1}$, we have effectively shown that $T(B_X)\supseteq (1-\epsilon )\tau T(B_Y)$, which immediately implies

$$\begin{aligned} q(T)\ge (1-\epsilon )^2\overline{q}(T). \end{aligned}$$

Therefore, $q(T)= \overline{q}(T)$.

Step 2: Consider $0<\tau < j(T)$.

By definition, $\left\| Tx \right\| \ge \tau \left\| x \right\|$ for all $x\in X$. For every $a\in B_{X'}$, the equation

$$\begin{aligned} \left\langle b_0,y \right\rangle =\left\langle a, T^{-1}y \right\rangle , \end{aligned}$$

defines a functional $b_0$ on $\text {ran}(T)$ with $\left\| b_0 \right\| \le \tau ^{-1}$. Choose an extension $b\in Y'$ such that $\left\| b \right\| \le \tau ^{-1}$. Consequently, we have $a=T'b$. Therefore, $T'(B_{Y'})\supseteq \tau B_{X'}$, and hence $q(T')\ge j(T)$.

Considering $0<\tau < q(T')$, for each $x\in X$, choose $\varphi \in B_{X'}$ such that $|\left\langle \varphi ,x \right\rangle |=\left\| x \right\|$. Since $T'(B_{Y'})\supseteq \tau B_{X'}$, we can find $b\in B_{Y'}$ with $T'b=\tau \varphi$. Consequently, we have

$$\begin{aligned} \left\| Tx \right\| \ge |\left\langle b,Tx \right\rangle |=|\left\langle T'b,x \right\rangle |=\left\langle \tau \varphi ,x \right\rangle =\tau \left\| x \right\| . \end{aligned}$$

This implies that $j(T)\ge q(T')$, and thus $j(T)=q(T')$. Similarly, one can prove that $j(T')\ge q(T)$.

Finally, let us consider $0<\tau <j(T')$. If there were $y\in B_Y$ such that $\tau y\not \in \overline{T(B_X)}$, then by the separation theorem, we could, in particular, find a functional $b\in Y'$ such that $|\left\langle b,\tau y \right\rangle |>1$ and $|\left\langle b, Tx \right\rangle |\le 1$ for all $x\in B_X$. This would imply that

$$\begin{aligned} \left\| T'b \right\| =\sup \{|\left\langle b, Tx \right\rangle |:x\in B_X\}\le 1< |\left\langle b,\tau y \right\rangle |\le \tau \left\| b \right\| , \end{aligned}$$

which is a contradiction. Thus, $\tau B_Y\subseteq \overline{T(B_X)}$. By the first step, we conclude that $q(T)\ge j(T')$. $\square$

Corollary 2.6

An operator $T\in B^R(X,Y)$ is a (metric) injection/surjection if and only if $T'\in B^R(Y',X')$ is a (metric) injection/surjection.

A right ${\mathbb {H}}$-Banach space Y possesses the extension property if for every injection $J\in B^R(X_0,X)$ and every operator $S_0\in B^R(X_0,Y)$ there exists an extension $S\in B^R(X,Y)$. In other words, the following diagram is commutative:

A stronger condition will be required in the sequel. The metric extension property means that for every metric injection $J\in B^R(X_0,X)$ and every operator $S_0\in B^R(X_0,Y)$, we can find $S\in B^R(X,Y)$ such that $S_0=SJ$ and $\left\| S \right\| =\left\| S_0 \right\|$. We will denote $X^{inj}=\ell _\infty (B_{X'})$ and $J_X:X\rightarrow {\mathbb {H}},\, x\mapsto (\left\langle \varphi ,x \right\rangle )$ where $\varphi \in B_{X'}$. Clearly $J_X$ is a metric injection from X into $X^{inj}$. Two examples of spaces that have a metric extension follow.

Lemma 2.7

[21, p. 33] $\ell _\infty$ has the metric extension property.

Lemma 2.8

[1, p. 84] Let $(\Omega ,\mu )$ be any measure space. Then $L_\infty (\Omega ,\mu )$ has the metric extension property.

Has shown in [1, p. 85], $L_\infty [0,1]$ is isormorphic to $\ell _\infty$. In this way every Banach space can be identified with a subspace of some Banach space having the metric extension property. More precisely,

Lemma 2.9

A Banach space has the extension property if and only if it is isomorphic to a complemented² subspace of some Banach space $\ell _\infty$.

To illustrate the concept of extension property it might be useful to look at spaces that don’t satisfy it. Indeed, as noted in [1, p. 86] there is no infinite-dimensional injective separable Banach spaces.

A Banach space Y has the lifting property if for each surjection $Q\in B^R(Y,Y_0)$ and all operators $S_0\in B^R(X,Y_0)$ there is a lifting $S\in B^R(X,Y)$. In other words, the following diagram commutes

The metric lifting property means that, given $\epsilon >0$, for each metric surjection $Q\in B^R(Y,Y_0)$ and every operator $S_0\in B^R(X,Y_0)$, we can find $S\in B^R(X,Y)$ such that $S_0=QS$ and $\left\| S \right\| =(1+\epsilon )\left\| S_0 \right\|$. We will denote $X^{sur}:=\ell _1(B_X)$ and $Q_X(\xi _x)=\sum _{B_X} \xi _xx$ for $(\xi _x)\in \ell _1(B_X)$, which is metric surjection from $X^{sur}$ onto X.

In analogy with the extension property, each Banach space can be identified with a quotient space of some Banach space having the metric lifting property.

3 Concept of s-Numbers in Quaternionic Analysis

3.1 Axiomatization of Quaternionic s-Number Theory and Initial Definitions

In the theory of s-numbers, an operator T is associated with various types of scalar sequences $s_n(T)$. In what follows, unless otherwise stated, all spaces are quaternionic right Banach spaces.

Based on [20], for $T\in B^R(X,Y)$, a mapping $s: T\rightarrow (s_n(T))_n$ will be referred to as an s-number function if it satisfies the following conditions:

A1.:

$\left\| T \right\| \ge s_1(T)\ge s_2(T)\ge \dots \ge 0$;

A2.:

$\forall n\in {\mathbb {N}}$, $s_n(S+T)\le s_n(S)+\left\| T \right\|$ for $X\overset{S,T}{\longrightarrow }\ Y$;

A3.:

$\forall n\in {\mathbb {N}}$, $s_n(BTA)\le \left\| B \right\| s_n(T)\left\| A \right\|$ for $X_0\overset{A}{\longrightarrow } X\overset{T}{\longrightarrow }\ Y\overset{B}{\longrightarrow }\ Y_0$;

A4.:

If $dim(X)\ge n$, $s_n(Id_X)=1$;

A5.:

If rank$(T)<n$ then $s_n(T)=0$.

For a fixed $n\in {\mathbb {N}}$ we call $s_n(T)$ the n-th s-number of T. According to the proposed axioms, s-number functions are continuous. Indeed, it immediately follows from ${\textbf {A2}}$ that $s_n(S)\le s_n(T)+ \left\| S-T \right\|$, which implies that

$$\begin{aligned} |s_n(S)-s_n(T)|\le |s_n(T)+ \left\| S-T \right\| -s_n(T)|=\left\| S-T \right\| . \end{aligned}$$

Also, the notions of multiplicity and additivity of s-numbers are relevant, which we adapt from [19, p. 327].

A2$^*$.:

If, for all $m,n\in {\mathbb {N}}$ and for all operators S and T, there holds

$$\begin{aligned} s_{n+m-1}(S+T)\le s_m(S)+s_n(T), \end{aligned}$$

then we say that the s-numbers are additive;

A3$^*$.:

If, for all $m,n\in {\mathbb {N}}$ and for all operators S and T, there holds

$$\begin{aligned} s_{n+m-1}(ST)\le s_m(S)s_n(T), \end{aligned}$$

then we say that the s-numbers are multiplicative.

Moreover, the dual s-number function can be introduced. For each s-number function we define the dual s-number function, $s'$, via $s_n'(T)=s_n(T')$ for all $T\in B^R(X,Y)$. An important remark must be made at this point: if we consider a right linear operator T acting between right Banach spaces, its dual will be a right linear operator acting between left Banach spaces. Consequently, the proposed axioms do not encompass this case. Therefore, when the notion of a dual s-number is required, it is necessary to additionally consider the axioms for a two-sided structure.

Additionally, following [21], an s-function, s, is called symmetric if, for each $n\in {\mathbb {N}}$, $s_n(T)\ge s_n(T')$. If equality is achieved, it is called completely symmetric.

In what follows we denote the evaluation map of X by $K_X$, i.e., the map defined as follows:

$$\begin{aligned} K_X: X \rightarrow X'', \quad x \mapsto K_x \end{aligned}$$

where $K_x:X'\rightarrow {\mathbb {H}}$ is given by $K_x(\varphi )=\left\langle \varphi ,x \right\rangle .$ It defines a right (left) functional on $X'$ if X is a right (left) Banach space. Finally, an s-function is said regular if, for each $n\in {\mathbb {N}}$, $s_n(T)=s_n(K_YT)$, for all $T\in B^R(X,Y)$.

3.2 The Particular Case of Hilbert Spaces

It has been well known for some time that s-numbers in the case of Hilbert spaces had various equivalent definitions, as observed in [13]. Later, in [20, pp. 203–204], it was demonstrated that the proposed axioms are consistent with this fact. We aim to generalize this result by showing that also in the quaternionic framework, the s-number function is unique, when considered over ${\mathbb {H}}$-Hilbert spaces, which we will denote, here and thereafter, by H.

To see this we require some technical results. Consider $T \in B^R(H)$, and let E be the spectral measure associated with the positive operator |T|, where $|T|=\sqrt{T^*T}$. Recall that in this case the spectral measure is reduced to the classic one, as observed in (2.8). In what follows, we denote

$$\begin{aligned} \sigma _n = \inf _{\sigma \ge 0} \left\{ \text {rank}(E(\sigma ,\infty )) < n\right\} . \end{aligned}$$

We ought to show that $s_n(T)=\sigma _n$. The following Lemma simplifies this task.

Lemma 3.1

Let $T\in B^R(H)$, then $s_n(T)=s_n(|T|)$.

Proof

From the polar representation of T (cf. Theorem 2.2), we know that there is a partial isometry U such that $T=U|T|$ and $|T|=U^*T$. The third axiom implies that

$$\begin{aligned} s_n(T)&\le \left\| U \right\| s_n(|T|) \le s_n(|T|)\le s_n(U^*T)\le \left\| U^* \right\| s_n(T)\le s_n(T) \end{aligned}$$

and thus $s_n(T)=s_n(|T|)$. $\square$

Lemma 3.2

rank$\left( \int _{\sigma _n+\epsilon }^\infty p E(dp) \right) <n$.

Proof

For any $\epsilon >0$ it follows by definition of $\sigma _n$ that

$$\begin{aligned} n>\text {rank}(E(\sigma _n+\epsilon ,\infty ))&=\text {rank}\left( \int _{\sigma _n+\epsilon }^\infty E(dp)\right) =\text {rank}\left( \int _{\sigma _n+\epsilon }^\infty pp^{-1}E(dp)\right) \\ {}&=\text {rank}\left( \int _{\sigma _n+\epsilon }^\infty pE(dp)\int _{\sigma _n+\epsilon }^\infty p^{-1}E(dp)\right) \\ {}&=\text {rank}\left( \int _{\sigma _n+\epsilon }^\infty pE(dp)\right) \text {rank}\left( \int _{\sigma _n+\epsilon }^\infty p^{-1}E(dp)\right) , \end{aligned}$$

which implies $\text {rank}\left( \int _{\sigma _n+\epsilon }^\infty p E(dp)\right) <n$. $\square$

Lemma 3.3

For any $\epsilon >0$ there holds that

$$\begin{aligned} E(\sigma _n-\epsilon ,\infty )=\left( \int _{0}^{\infty } p E(dp)\right) \left( \int _{\sigma _n-\epsilon }^{\infty }p^{-1} E(dp)\right) \end{aligned}$$

Proof

$$\begin{aligned} \int _{0}^{\infty } p&E(dp)\int _{\sigma _n-\epsilon }^{\infty } p^{-1} E(dp)\\ {}&=\left( \int _{0}^{\sigma _n-\epsilon }p E(dp)+\int _{\sigma _n-\epsilon }^{\infty }p E(dp)\right) \left( \int _{\sigma _n-\epsilon }^{\infty }p^{-1} E(dp)\right) \\ {}&=\int _{0}^{\sigma _n-\epsilon }p E(dp)\int _{\sigma _n-\epsilon }^{\infty }p^{-1} E(dp)+\int _{\sigma _n-\epsilon }^{\infty }p E(dp)\int _{\sigma _n-\epsilon }^{\infty }p^{-1} E(dp)\\&=\int _0^\infty p \chi _{(0,\sigma _n-\epsilon )}E(dp)\int _0^\infty p^{-1} \chi _{(\sigma _n-\epsilon ,\infty )}E(dp)+\int _{\sigma _n-\epsilon }^{\infty }pp^{-1} E(dp)\\&=\int _0^\infty \underbrace{p \chi _{(0,\sigma _n-\epsilon )}p^{-1} \chi _{(\sigma _n-\epsilon ,\infty )}}_{=0}E(dp)+\int _{\sigma _n-\epsilon }^ {\infty }E(dp)\\ {}&=E(\sigma _n-\epsilon ,\infty ). \end{aligned}$$

$\square$

Theorem 3.4

For $S\in B^R(H)$ there holds $s_n(T)=\sigma _n$.

Proof

Considering an $\epsilon >0$, then we can write

$$\begin{aligned} |S|=\int _{0}^{\infty }p E(dp)=\int _{0}^{\sigma _n+\epsilon }p E(dp)+\int _{\sigma _n+\epsilon }^{\infty }p E(dp). \end{aligned}$$

As an application of A5., it follows from Lemma 3.2 that

$$\begin{aligned} s_n(|S|)\le \left\| \int _{0}^{\sigma _n+\epsilon }p E(dp) \right\| +s_n\left( \int _{\sigma _n+\epsilon }^{\infty }p E(dp)\right) \le \sigma _n+\epsilon \end{aligned}$$

because

$$\begin{aligned} \left\| \int _0^{\sigma _n+\epsilon }p E(dp) \right\| \le \left\| Id \right\| _\infty \left\| E(0,\sigma _n+\epsilon ) \right\| \le \left\| E \right\| (\sigma _n+\epsilon )=\sigma _n+\epsilon , \end{aligned}$$

where $\left\| Id \right\| _\infty$ denotes the supremum norm of the identity operator.

Now we consider $0<\epsilon <\sigma _n$. By Lemma 3.3

$$\begin{aligned} E(\sigma _n-\epsilon ,\infty )=\left( \int _{0}^{\infty }p E(dp)\right) \left( \int _{\sigma _n-\epsilon }^{\infty }p^{-1} E(dp)\right) \end{aligned}$$

By definition of $\sigma _n$, rank$\left( E(\sigma _n-\epsilon ,\infty )\right) \ge n$. Since $|S|=\int _{0}^{\infty }p E(dp)$, it follows from the third axiom of s-number theory that

$$\begin{aligned} 1&=s_n(E(\sigma _n-\epsilon ,\infty ))\le s_n(|S|)\left\| \int _{\sigma _n-\epsilon }^{\infty }p^{-1} E(dp) \right\| \le s_n(|S|) \sup _{p\in (\sigma _n-\epsilon ,\infty )} p^{-1}\\ {}&= s_n(|S|)(\sigma _n-\epsilon )^{-1}. \end{aligned}$$

Therefore $\sigma _n-\epsilon \le s_n(|S|)\le \sigma _n+\epsilon$ for all $\epsilon >0$. $\square$

As a consequence, according to the “classic” definition of s-numbers,

Corollary 3.5

Consider a compact right linear operator acting between right ${\mathbb {H}}$-Hilbert spaces, S. Then $s_n(S)$ is the n-th eigenvalue of |S|.

As a consequence, s-numbers on quaternionic Hilbert spaces possess the properties of additivity and multiplicativity. Indeed, the same ideas as the ones presented in [12, pp. 764–765] can be used.

This contrasts with the case of general Banach spaces. More precisely, only on Hilbert spaces can one use Riesz representation Theorem, so that the operator E in (2.4) is well defined. Therefore, a priori, one should not expect the uniqueness claim to hold in a general Banach space.

This is indeed the case. In fact, there are several examples of s-number functions when considered over Banach spaces, as we will see in the subsequent section.

3.3 Examples of s-Number Functions on Quaternionic Banach Spaces

In what follows, let X and Y denote right ${\mathbb {H}}$-Banach spaces and consider $T\in B^R(X,Y)$.

• Approximation numbers: First introduced in [20, p. 204], the n-th approximation number of T is defined by

  $$\begin{aligned} a_n(T):=\inf \{\left\| T-A \right\| \}, \end{aligned}$$

  where the infimum is taken over all right linear and bounded operators A that have a rank$(A)<n$. Thus, one is essentially measuring how well can the operator T be approximated by operators of finite rank.

• Gelfand numbers: Adapting the definition provided in [20, p. 206], the n-th Gelfand number of T is defined as follows:

  $$\begin{aligned} c_n(T):=\inf \{\left\| TJ_M^X \right\| \}, \end{aligned}$$

  where the infimum is taken over all subspaces $M\subset X$ with codimension smaller than n. Hence, one is essentially measuring how much does the norm of the operator T change according to the change of its domain.

• Kolmogorov numbers: The natural counterpart of Gelfand numbers, as we will see in Theorem 3.9, are the Kolmogorov numbers. Following classic Kolmogorov numbers, introduced in [20, p. 208], the n-th Kolmogorov number of T is defined as follows:

  $$\begin{aligned} d_n(T):=\inf \{\left\| Q^Y_N T \right\| \} \end{aligned}$$

  where the infimum is taken over all subspaces $N\subset Y$, that have dimension smaller then n. One readily observes that the variation occurs on the image of the operator rather than its domain.

• Hilbert numbers: Equivalent to measuring how isomorphic the domain and the image of an operator are (cf. [7, pp. 35–36]), the classic Hilbert numbers were studied in [2], where we take inspiration from. Consider a right ${\mathbb {H}}$-Hilbert space, H. The n-th Hilbert number of T is defined as follows:

  $$\begin{aligned} h_n(T)=\sup \{s_n(BTA)\}, \end{aligned}$$

  where the supremum is taken over all right linear and bounded operators $A:H\rightarrow X$ and $B:Y\rightarrow H$ for which $\left\| A \right\| ,\left\| B \right\| \le 1$. It is a consequence of Theorem 3.4 that the choice of s-number function will not alter value of $s_n(BTA)$, since the operator BTA is acting between Hilbert spaces.

• Weyl numbers: The classic Weyl numbers were introduced in [22, p. 149], which serves as a foundation for our considerations. The n-th Weyl number of T is given by

  $$\begin{aligned} x_n(T)=\sup a_n(TA), \end{aligned}$$

  where the supremum is taken over all $A\in B^R(\ell _2, X)$ for which $\left\| A \right\| \le 1$.

• Chang numbers: First introduced in [19, p. 330]. The $n-th$ Chang number of T is defined as follows,

  $$\begin{aligned} y_n(T)=\sup a_n(BT), \end{aligned}$$

  where the supremum is taken over all $B\in B^R(Y, \ell _2)$ for which $\left\| B \right\| \le 1$.

We would like to remark that, all the presented s-number functions are additive and multiplicative. Moreover, all the so far presented s-numbers are a direct adaption of the works cited. Next we introduce a new concept of s-numbers which might shed some light on the theory of the trace of a quaternionic operator.

3.4 Nuclear Numbers

The notion of nuclear operator goes back to the works of Grothendieck in [14]. In a concise manner, these are the operators acting between (complex) Banach spaces for which a notion of the trace of an operator is well defined. Hereby, a nuclear operator is linear and bounded operator $T:X\rightarrow X$ such that there are $(x'_i)\in X'$ and $(y_i)\in Y$ for which

$$\begin{aligned} T=\sum _{i=1}^\infty x'_i\otimes y_i\qquad \text {and}\qquad \sum _{i=1}^\infty \left\| x'_i \right\| \left\| y_i \right\| <\infty . \end{aligned}$$

To the former term we refer to as nuclear representation of the operator T. We equip it with the norm

$$\begin{aligned} \left\| T \right\| =\inf \sum _{i=1}^\infty \left\| x'_i \right\| \left\| y_i \right\| , \end{aligned}$$

where the infimum is taken over all nuclear representations of T.

This concept was later generalized by Pietsch in [21, p. 243] to (r,p,q)-nuclear operators as follows: let $0<r\le \infty$, $1\le p,q\le \infty$ and $1+\frac{1}{r}\ge \frac{1}{p}+\frac{1}{q}$, for a sequence $(\sigma _i)\in \ell _r$, $(x'_i)\in w_{q'}(X')$ and $(y_i)\in w_{p'}(Y)$ a linear and bounded operator $T: X\rightarrow Y$ is said (r,p,q)-nuclear operator if

$$\begin{aligned} T=\sum _{i=1}^\infty \sigma _i x_i'\otimes y_i\qquad \text {and}\qquad \sum _{i=1}^\infty \left\| \sigma _i|\ell _r \right\| \left\| x'_i|w_{q'}(X') \right\| \left\| y_i|w_{p'}(Y) \right\| <\infty . \end{aligned}$$

(3.1)

The set of these operators shall be denoted by ${\mathfrak {R}}_{r,p,q}(X, Y)$. Moreover, the former term in (3.1) is referred to as (r,p,q)-nuclear representation of the operator T. We equip it with the norm

$$\begin{aligned} \left\| T|{\mathfrak {R}}_{r,p,q} \right\| :=\inf \sum _{i=1}^\infty \left\| \sigma _i|\ell _r \right\| \left\| x'_i|w_{q'}(X') \right\| \left\| y_i|w_{p'}(Y) \right\| , \end{aligned}$$

where the infimum is taken over all (r,p,q)-nuclear representations of T. As noted in [21, p. 382], for a Hilbert space H, ${\mathfrak {R}}_{1,1,2}(H,H)={\mathfrak {S}}_1(H)$. These operators play an important role into the generalization of Grothendieck-Lidskii trace formula. It states for a given operator T there holds

$$\begin{aligned} Tr(T)=\sum _{i=1}^\infty \lambda _i(T), \end{aligned}$$

where $(\lambda _i(T))$ is the sequence of eigenvalues of T.

Indeed Grothendieck-Lidskii trace formula appears as a special case for nuclear operators. This is explained in detail in [21, pp. 373–382]. Essentially, for $T\in {\mathfrak {R}}_{1,1,2}(H,H)$ one can show that

$$\begin{aligned} \text {trace}(T)=\sum _{i=1}^\infty \sigma _i\left\langle x_i,a_i \right\rangle \end{aligned}$$

for any (1, 1, 2)- nuclear representation $T=\sum _{i=1}^\infty \sigma _i x_i\otimes a_i$. Moreover, since ${\mathfrak {R}}_{\frac{2}{3},1,1}\subseteq {\mathfrak {R}}_{1,1,2}$ the result also holds for $(\frac{2}{3},1,1)$. However, for $\frac{2}{3}<r\le 1$ one cannot expect a trace formula to hold. In fact, counter examples have already been found, cf. [21, p. 138]. It is clear that these concepts can be directly generalized for right linear operators acting between right Banach spaces. For the sake of completeness we present the following definition, adapted from the aforementioned discussion.

Definition 3.6

[Quaternionic (r,p,q)-nuclear operators] Let X and Y be ${\mathbb {H}}$-Banach spaces and consider $T\in B^R(X,Y)$. Moreover, for $\varphi \in X'$ and $y\in Y$, let $\varphi \otimes y$ denote the map

$$\begin{aligned} \varphi \otimes y:\, X\ni x\longmapsto y\left\langle \varphi ,x \right\rangle . \end{aligned}$$

Then, T will be referred to as a (right) (r,p,q)-nuclear operator if

$$\begin{aligned} T=\sum _{i=1}^\infty \sigma _i x_i'\otimes y_i\qquad \text {and}\qquad \sum _{i=1}^\infty \left\| \sigma _i|\ell _r \right\| \left\| x'_i|w_{q'}(X') \right\| \left\| y_i|w_{p'}(Y) \right\| <\infty . \end{aligned}$$

The same notation for the norm will be used.

Recently, there has been a growing interest in extending the formula mentioned above for quaternionic cases (cf. [11]). Motivated by the natural extension of s-numbers to quaternionic space and the significance of nuclear operators in the study of the Grothendieck-Lidskii trace formula, we introduce the concept of nuclear numbers. Let $T\in B^R(X,Y)$ be a (r, p, q)-nuclear operator. Consider the sequences $(\sigma _n)$, $(x'_n)$ and $(y_n)$ for which

$$\begin{aligned} \sum _{i=1}^\infty \left\| \sigma _n|\ell _r \right\| \left\| x'_n|w_{q'}(X') \right\| \left\| y_n|w_{p'}(Y) \right\| =\left\| T|{\mathfrak {R}}_{r,p,q} \right\| . \end{aligned}$$

Then we define the n-th nuclear number of T as

$$\begin{aligned} g_n(T)=\left\| \sigma _n^*|\ell _r \right\| \left\| (x_n')^*|w_{q'}(X') \right\| \left\| y_n^*|w_{p'}(Y) \right\| , \end{aligned}$$

where the $^*$ notation represents the decreasing rearrangement of the corresponding sequence.

Remark

As the (r, p, q)-norm is defined as an infimum, it might happen that the actual sequences to define the n-nuclear number do not exist. In this case, given that s-number functions are continuous, we can define the nuclear numbers as the limit of sequences that converge to the desired sequences.

Theorem 3.7

Let $T\in B^R(X,Y)$. Then $nuc:T\rightarrow (g_n(T))_n$ is an additive s-number function.

Proof

To prove that $g_n(BTA)\le \left\| B \right\| g_n(T)\left\| A \right\|$ for suitable suitable operators A, B and T, suppose that the (r, p, q)-nuclear representation of T is given by $\sum _{i=1}^\infty \sigma _i x_i\otimes y_i$. Then the (r, p, q)-nuclear representation of BTA is given by $\sum _{i=1}^\infty \sigma _i A'x_i\otimes By_i$. Therefore

$$\begin{aligned} g_n(BTA)&= \left\| \sigma _i^*|\ell _r \right\| \left\| A'x_i^*|w_{q'}(X') \right\| \left\| By_i^*|w_{p'}(Y) \right\| \\ {}&\le \left\| \sigma _i^*|\ell _r \right\| \left\| A \right\| \left\| x_i^*|w_{q'}(X') \right\| \left\| B \right\| \left\| y_i^*|w_{p'}(Y) \right\| =\left\| B \right\| g_n(T)\left\| A \right\| \end{aligned}$$

To show that $g_n(T)=0$ when rank$(T)<n$, without loss of generality let rank$(T)=n-1$. Then $T=\sum _{i=1}^{n-1}\sigma _i x_i\otimes y_i$. We can technically extend this sum to n since after the $(n-1)$-th term we are just adding zeros. It immediately follows that $s_n(T)=0$.

The remaining properties follow immediately. $\square$

3.5 Other Properties

More properties can be extracted from the so far discussed s-numbers. The table bellow resumes some of their properties. The proofs can be found in [7].

s-Numbers	Complete symmetry	Special attributes
$a_n(T)$	For compact operators	Largest s-number function
$c_n(T)$	No	Largest injective s-number function
$d_n(T)$	No	Largest surjective s-number function
$h_n(T)$	Yes	Smallest s-number function
$x_n(T)$	No	–
$y_n(T)$	No	–

An important tool for proving the complete symmetry results is what is known as the Principle of local reflexivity. In the quaternionic setting it can be stated as follows: for a right ${\mathbb {H}}$-Banach space X and for a finite dimensional subset $U\subset X''$, if $\epsilon > 0$, then there exists a one-to-one operator $T: U\rightarrow X$ with $T(x)=x$ for all $x\in U\cap X$ and $\left\| T \right\| \left\| T^{-1} \right\| <1+\epsilon$. More details can be found in [7, p. 30].

3.6 Relations Between s-Numbers

Duality relations can be established between different s-number functions as the next theorems show, that explain the absence of the aforementioned complete symmetry of some s-numbers. Moreover, some s-numbers play an interesting role in classifying compact operators, as we will see. First we require some preliminary results. The classic results can be found in [20, p. 205] and [20, pp. 211–212], respectively.

Theorem 3.8

Let X and Y be right ${\mathbb {H}}$-Banach Spaces. If $T\in B^R(X,Y)$ and additionally Y has the metric extension property then $c_n(T)=a_n(T)$. Moreover, if additionally X has the metric lifting property $d_n(T)=a_n(T)$.

Proof

Assume that Y has the metric extension property. Take $S\in B^R(X,Y)$. It suffices to show that $a_n(S)\le c_n(S)$. Considering $\epsilon >0$ we can choose a subspace M of X such that

$$\begin{aligned} \left\| SJ^X_M \right\| \le c_n(S)+\epsilon \end{aligned}$$

for which codim$(M)<n$. As Y has the metric extension property, there is an extension $T\in B^R(X,Y)$ of $SJ^X_M$ for which $\left\| T \right\| =\left\| SJ^X_M \right\|$. Set $A=S-T$. By definition, for all $x\in M$, $Ax=0$, therefore rank$(A)<n$. Hence,

$$\begin{aligned} a_n(S)\le \left\| S-A \right\| =\left\| T \right\| =\left\| SJ^X_M \right\| \le c_n(S)+\epsilon . \end{aligned}$$

Now assume that X has the metric lifting property. Take $S\in B^R(X,Y)$. Again it suffices to show that $a_n(S)\le d_n(S)$. Considering $\epsilon >0$ we can choose a subspace N of Y such that

$$\begin{aligned} \left\| Q^Y_NS \right\| \le d_n(S)+\epsilon \end{aligned}$$

for which dim$(N)<n$. As X has the metric lifting property, there is a lifting $T\in B^R(X,Y)$ of $Q^Y_NS$ for which $\left\| T \right\| =(1+\epsilon )\left\| Q^Y_NS \right\|$. Set $A=S-T$. By definition, for all $x\in X$, $Ax\in N$, therefore rank$(A)<n$. Therefore

$$\begin{aligned} a_n(S)\le \left\| S-A \right\| =\left\| T \right\| =(1+\epsilon )\left\| Q^Y_NS \right\| \le (1+\epsilon )d_n(S). \end{aligned}$$

$\square$

Consequently, duality relations between Gelfand and Kolmogorov numbers can be established as follows.

Theorem 3.9

Consider a right linear and bounded operator, acting between right ${\mathbb {H}}$-Banach spaces, $T:X\rightarrow Y$. Then $c_n(T)=d_n(T')$ and $d_n(T)\ge c_n(T')$. Additionally, if the operator is compact, then $d_n(T)=c_n(T')$.

Proof

Since $J_Y$ is a metric injection it follows from Corollary 2.6 that $J'_Y$ is a metric surjection. Thus, by the surjectivity of Kolmogorov numbers and the symmetry of approximation numbers, we have

$$\begin{aligned} d_n(T')=d_n(T'J_Y')\le a_n(T'J_Y')\le a_n(J_YT)=c_n(T). \end{aligned}$$

Analogously from Corollary 2.6, $Q'_X$ is a metric injection which allows us to show that $c_n(T')\le d_n(T)$. An adaptation from the proof found in [21, p. 153] proves that these are regular s-numbers. Therefore,

$$\begin{aligned} c_n(T)=c_n(K_YT)=c_n(T''K_X)\le c_n(T'')\le d_n(T'), \end{aligned}$$

since $K_YTJ_M^X=T''K_XJ_M^X$. This follows from the fact that for each $x\in M$, $T''(K_x)=K_{Tx}$.

Now assume that the operator is compact. By definition of $X^{sur}$ and by Lemma 2.9 it follows that $(X^{sur})'$ has the metric extension property. From the complete symmetry of approximation numbers and with Theorem 3.8 we have

$$\begin{aligned} d_n(T)=a_n(TQ_X)=a_n(Q'_XT')=c_n(Q'_XT')\le c_n(T'). \end{aligned}$$

$\square$

Theorem 3.10

[18, p. 90] For a right ${\mathbb {H}}$-Hilbert space H and a right ${\mathbb {H}}$-Banach space Y, if $T\in B^R(H,Y)$, then $c_n(T)=a_n(T)$. On the other hand if $T\in B^R(X,H)$, for a right ${\mathbb {H}}$-Banach space X, there holds $d_n(T)=a_n(T)$.

Proof

We only show the first claim as the second one follows the same lines. It remains to show that $a_n(T)\le c_n(T)$ for any n, given that approximation numbers are the largest s-number function. Consider a n-codimensional subspace of H, M. Let $P\in B^R(H)$ denote the orthogonal projection from H onto M and set $L=T(Id-P)$. As $M\subseteq \ker (L)$ it follows that

$$\begin{aligned} \text {rank}(L)=\text {codim}(\ker (L))\le \text {codim}(M)=n. \end{aligned}$$

Therefore,

$$\begin{aligned} a_n(T)\le \left\| T-L \right\| =\left\| TP \right\| =\left\| TJ^H_M \right\| . \end{aligned}$$

Since M is arbitrary the claim follows. $\square$

As a consequence of 3.10 and the duality relations previously discussed we have the following theorem.

Theorem 3.11

[19, p. 329] Consider a right linear and bounded operator acting between right ${\mathbb {H}}$-Banach Spaces, T. Then $x_n(T')=y_n(T)$ and $y_n(T')=x_n(T)$.

Proof

Consider a sequence $(A_m)_{m\in {\mathbb {N}}}\in B^R(\ell _2,X)$ such that for each $m\in {\mathbb {N}}$, $\left\| A_m \right\| \le 1$ and $a_n(TA_m)\rightarrow \sup \left\{ a_n(TA): \left\| A \right\| \le 1\right\}$, as $m\rightarrow \infty$. On the one hand, from the definition of Weyl number it holds

$$\begin{aligned} a_n(TA_m)\rightarrow x_n(T),\quad \text {as } m\rightarrow \infty . \end{aligned}$$

On the other hand, it follows from Theorem 3.10 that

$$\begin{aligned} a_n(TA_m)=c_n(TA_m)=d_n(A_m'T')=a_n(A_m'T'). \end{aligned}$$

The definition of transpose operator implies that

$$\begin{aligned} a_n(A_m'T')\rightarrow \sup \left\{ a_n(A'T'),\quad A':X'\rightarrow \ell _2,\, \left\| A' \right\| \le 1\right\} =y_n(T'). \end{aligned}$$

The uniqueness of limit gives $x_n(T)=y_n(T')$. An analogous proof proves the other equality. $\square$

Both Gelfand numbers and Kolmogorov numbers describe compact operators in a one to one fashion as we will see in the following theorem.

Theorem 3.12

A right linear operator T is compact if and only if $(c_n(T))_n\rightarrow 0$.

Proof

Consider a right compact operator T acting between right Banach spaces X and Y. Then, for each $\epsilon >0$, there are $y_1, \dots , y_n\in Y$ such that the set

$$\begin{aligned} \left\{ y_i+\epsilon B_Y: 1\le i\le n\right\} \quad \text {is a finite cover of } T(B_X). \end{aligned}$$

For $i=1,\dots n$, consider $b_i\in B_{Y'}$ for which $|\left\langle b_i,y_i \right\rangle |=\left\| y_i \right\|$ and set

$$\begin{aligned} M:=\{x\in X: \left\langle b_j, Tx \right\rangle =0 \text { for } i=1,\dots n\}. \end{aligned}$$

If $x\in M\cap B_X$ then we can take $y_j$ such that $\left\| Tx -y_j \right\| \le \epsilon$ and since

$$\begin{aligned} \left\| y_j \right\| =|\left\langle b_j, y_j \right\rangle |\le |\left\langle b_j, y_j-Tx \right\rangle |+|\left\langle b_j, Tx \right\rangle |\le \left\| y_j-Tx \right\| \left\| b_j \right\| \le \epsilon \end{aligned}$$

we conclude that

$$\begin{aligned} \left\| Tx \right\| \le \left\| Tx-y_j \right\| +\left\| y_j \right\| \le 2\epsilon , \end{aligned}$$

which implies $\left\| TJ_M^X \right\| \le 2\epsilon$ because we considered $x\in M\cap B_X$. This gives us the desired claim since $c_n(T)\le \left\| TJ_M^X \right\|$. Conversely, assume that $c_n(T)\rightarrow 0$. Then, for each $\epsilon >0$, there is a finite codimensional M for which $\left\| TJ_M^X \right\| \le \epsilon$. Let N be such that $X=M\oplus N$ and denote the corresponding projections by $P_N$ and $P_M$. Set $\delta =\min \left( \frac{\epsilon }{\left\| T \right\| },1\right)$. Since $P_N$ is of finite rank, there are $x_1,\dots x_m\in B_X$ such that

$$\begin{aligned} \{P_Nx_i+\delta B_Y\}_{i=1}^m \text { covers } P_N(B_X). \end{aligned}$$

Therefore, for $x\in B_X$, we can choose $x_i$ with $\left\| P_Nx-P_Nx_i \right\| \le \delta$. Hence, it follows that

$$\begin{aligned} \left\| P_Mx-P_Mx_i \right\| \le \left\| x-x_i \right\| +\left\| P_Nx-P_Nx_i \right\| \le 2+\delta \le 3 \end{aligned}$$

and so

$$\begin{aligned} \left\| T(x-x_i) \right\| \le \left\| TP_M(x-x_i) \right\| +\left\| TP_N(x-x_i) \right\| \le 3\left\| TJ^X_M \right\| +\delta \left\| T \right\| \le 4 \epsilon . \end{aligned}$$

Since $\epsilon$ is arbitrary we have therefore found a finite subcover of $T(B_X)$, more precisely being given by

$$\begin{aligned} T(B_X)=\bigcup _{i=1}^n\{Tx_i+4\epsilon B_Y\}. \end{aligned}$$

$\square$

As a consequence of Theorems 3.9 and 3.12 and the symmetry of Kolmogorov and Gelfand numbers we conclude that also Kolmogorov numbers describe compact operators as well. Indeed,

Theorem 3.13

A right linear operator T is compact if and only if $(d_n(T))_n\rightarrow 0$.

Finally, we can establish a relation of order. The arrows point from the larger s-numbers to the smaller:

These relations will play an important role in the classification of the Schatten classes defined in the next section. The converse inequalities depend on constants that depend on the order of the s-number, cf. [7, p. 44].

4 Quaternionic Schatten Classes

Following the ideas proposed by Pietsch, we can introduce what is known as Schatten classes of operators, but in this case, acting on right ${\mathbb {H}}$-Banach spaces, with respect to a given s-number function, which will be denoted by ${\mathfrak {S}}^{(s)}_p$.

In the particular case of quaternionic Hilbert spaces, the uniqueness of s-numbers obtained in Theorem 3.4, implies that this definition yields to a unique classification of compact operators as follows: For right ${\mathbb {H}}$-Hilbert spaces X and Y, let $\sigma _n(T)=\inf _{\sigma }\{\text {rank}(E(\sigma ,\infty ))<n\}$ where E is the spectral measure associated to the operator |T|, then

$$\begin{aligned} {\mathfrak {S}}_p=\left\{ T\in B^R(X,Y): \left\| (\sigma _n(T))_n|\ell _p \right\| <\infty \right\} . \end{aligned}$$

We emphasize that this definition coincides with the classical definition of Schatten classes. Consequently, the values $\sigma _n(T)$ correspond to singular values appearing in the singular value decomposition of the operator T, as noted in Corollary 3.5. This differs from previously proposed definitions such as the one presented in [4]. There, Schatten classes were essentially defined with respect to a given slice (more precisely with respect to a anti-selfadjoint unitary operator, J). However, this approach has a drawback: basis dependence. With the proposed axiomatization, this particular issue is solved.

In the case of right ${\mathbb {H}}$-Banach spaces we then have several notions of Schatten classes, each of which associated to a specific s-number function. For instance, one can refer to the approximation-Schatten class of an operator as

$$\begin{aligned} {\mathfrak {S}}^{(a)}_p=\left\{ T\in B^R(X,Y): \left\| a_n(T)|\ell _p \right\| <\infty \right\} . \end{aligned}$$

The independence of the basis of the s-number function seen so far, yields the desired basis independence. More precisely we have the following definition.

Definition 4.1

Consider right ${\mathbb {H}}$-Banach spaces X and Y and $T\in B^R(X,Y)$. For $1\le p<\infty$ we denote,

$$\begin{aligned} {\mathfrak {S}}^{(s)}_p=\left\{ T\in B^R(X,Y): \left\| s_n(T)|\ell _p \right\| <\infty \right\} , \end{aligned}$$

and for $p=\infty$

$$\begin{aligned} {\mathfrak {S}}^{(s)}_\infty =\left\{ T\in B^R(X,Y): \lim \limits _{n\rightarrow \infty } s_n(T) =0\right\} . \end{aligned}$$

These spaces are equipped with the quasi-norm

$$\begin{aligned} \left\| T|{\mathfrak {S}}^{(s)}_p \right\| =\left\| s_n(T)|\ell _p \right\| . \end{aligned}$$

According to this definition, it is then a consequence of Theorems 3.12 and 3.13 that ${\mathfrak {S}}^{(c)}_\infty ={\mathfrak {S}}^{(d)}_\infty =K^R$, the set of compact right linear operators.

Let $T\in {\mathfrak {S}}_{p}^{(s)}$. Recall that for any $x\in B_X$ and $\varphi \in B_{Y'}$

$$\begin{aligned} \varphi \otimes Tx: Y\rightarrow Y,\quad y\mapsto Tx \left\langle \varphi , y \right\rangle . \end{aligned}$$

In this case, if $1'\in X'$ denotes the functional on X that maps to 1 we have $x=x\left\langle 1',\cdot \right\rangle =1'\otimes x$. Moreover, observe that $\varphi \otimes 1=\left\langle \varphi ,\cdot \right\rangle$. With a slight abuse of notation we can therefore write

$$\begin{aligned} \left\langle \varphi ,Tx \right\rangle =\left\langle \varphi ,\cdot \right\rangle Tx\left\langle 1',\cdot \right\rangle =\varphi \otimes 1T1'\otimes x. \end{aligned}$$

This serves to show that

$$\begin{aligned} |\left\langle \varphi ,Tx \right\rangle |\le \underbrace{\left\| \varphi \otimes 1 \right\| }_{\le 1} \left\| T|{\mathfrak {S}}_{p}^{(s)} \right\| \underbrace{\left\| 1'\otimes x \right\| }_{\le 1} \le \left\| T|{\mathfrak {S}}_{p}^{(s)} \right\| . \end{aligned}$$

holds. This implies that $\left\| T \right\| \le \left\| T|{\mathfrak {S}}_{p}^{(s)} \right\|$.

Consequently, if one considers a Cauchy sequence $(T_k)\in {\mathfrak {S}}_{p}^{(s)}$, there exists a limit T with the respect to the operator topology because $\left\| T_k-T_h \right\| \le \left\| T_k-T_h|{\mathfrak {S}}_{p}^{(s)} \right\|$. From the continuity of s-number functions it follows that $s_n(T_k-T_h)\rightarrow 0$ as $h,k\rightarrow \infty$, therefore $\left\| T_k-T_h|{\mathfrak {S}}_{p}^{(s)} \right\| \rightarrow 0$. This proves the following

Theorem 4.2

For any s-number function and for any $1\le p\le \infty$, ${\mathfrak {S}}_{p}^{(s)}$ is a right quasi-Banach space.

The order of relation previously established for the different s-numbers on Banach spaces allows us to obtain the following diagram of embeddings (which are sharp) between the several Schatten classes:

For more details see [7, p. 44].