The new class $L_{z,p,E}$ of $s-$ type operators

1.   Introduction

In this study, the set of all natural numbers is represented by $\mathbb{N}$ and the set of all nonnegative real numbers is represented by $\mathbb{R} ^{+}$.

If the dimension of the range space of a bounded linear operator is finite, it is called a finite rank operator ^[1].

Throughout this study, $X$ and $Y$ denote real or complex Banach spaces. The space of all bounded linear operators from $X$ to $Y$ is denoted by $\mathcal{B}\left(X, Y\right)$ and the space of all bounded linear operators from an arbitrary Banach space to another arbitrary Banach space is denoted by $\mathcal{B}$.

The theory of operator ideals is a very important field in functional analysis. The theory of normed operator ideals first appeared in 1950's in ^[2]. In functional analysis, many operator ideals are constructed via different scalar sequence spaces. An $s$- number sequence is one of the most important examples of this. The definition of $s$- numbers goes back to E. Schmidt ^[3], who used this concept in the theory of non-selfadjoint integral equations. In Banach spaces there are many different possibilities of defining some equivalents of $s$- numbers, namely Kolmogorov numbers, Gelfand numbers, approximation numbers, and several others. In the following years, Pietsch give the notion of $s$- number sequence to combine all $s$- numbers in one definition ^{[4, 5, 6]}.

A map

which assigns a non-negative scalar sequence to each operator is called an $s$-number sequence if for all Banach spaces $X, Y, X_{0}$ and $Y_{0}$ the following conditions are satisfied:

($i$) $\left \Vert K\right \Vert = s_{1}\left(K\right) \geq s_{2}\left(K\right) \geq \ldots \geq 0$, for every $K\in \mathcal{B} \left(X, Y\right),$

($ii$) $s_{p+r-1}\left(L+K\right) \leq s_{p}\left(L\right) +s_{r}\left(K\right)$ for every $L, K\in \mathcal{B}\left(X, Y\right)$ and $p, r\in \mathbb{N},$

($iii$) $s_{r}\left(MLK\right) \leq \left \Vert M\right \Vert s_{r}\left(L\right) \left \Vert K\right \Vert$ for all $M\in \mathcal{B} \left(Y, Y_{0}\right),$ $L\in \mathcal{B}\left(X, Y\right)$ and $\ K\in \mathcal{B}\left(X_{0}, X\right),$ where $X_{0}, Y_{0}$ are arbitrary Banach spaces,

($iv$) If $rank\left(K\right) \leq r,$ then $s_{r}\left(K\right) = 0,$

($v$) $s_{n-1}\left(I_{n}\right) = 1,$ where $I_{n}$ is the identity map of $n$-dimensional Hilbert space $l_{2}^{n}$ to itself ^[7].

$s_{r}\left(K\right)$ denotes the $r-th$ $s-$number of the operator $K$.

Approximation numbers are frequently used examples of s-number sequence which is defined by Pietsch. $a_{r}\left(K\right)$, the $r$-th approximation number of a bounded linear operator is defined as

where $K\in \mathcal{B}\left(X, Y\right)$ and $r\in \mathbb{N}$ ^[4]. Let $K\in \mathcal{B}\left(X, Y\right)$ and $r\in \mathbb{N}.$ The other examples of s-number sequences are given in the following, namely $Gel'fand$ number $\left(c_{r}\left(K\right) \right)$, $Kolmogorov$ number $\left(d_{r}\left(K\right) \right)$, $Weyl$ number $\left(x_{r}\left(K\right) \right)$, $Chang$ number $\left(y_{r}\left(K\right) \right),$ $Hilbert$ number $\left(h_{r}\left(K\right) \right),$ etc. For the definitions of these sequences we refer to ^[1].

In the sequel there are some properties of $s-$number sequences.

When any isometric embedding $\mathcal{J}\in \mathcal{B}\left(Y, Y_{0}\right)$ is given and an s-number sequence $s = \left(s_{r}\right)$ satisfies $s_{r}\left(K\right) = s_{r}\left(\mathcal{J}K\right)$ for all $K\in \mathcal{B}\left(X, Y\right)$ the s-number sequence is called injective [8,p.90].

Proposition 1. [8,p.90-94] The number sequences $\left(c_{r}\left(K\right) \right)$ and $\left(x_{r}\left(K\right) \right)$ are injective.

When any quotient map $\mathcal{S}\in \mathcal{B}\left(X_{0}, X\right)$ is given and an s-number sequence $s = \left(s_{r}\right)$ satisfies $s_{r}\left(K\right) = s_{r}\left(K\mathcal{S}\right)$ for all $K\in \mathcal{B}\left(X, Y\right)$ the s-number sequence is called surjective [8,p.95].

Proposition 2. [8,p.95] The number sequences $\left(d_{r}\left(K\right) \right)$ and $\left(y_{r}\left(K\right) \right)$ are surjective.

Proposition 3. [8,p.115] Let $K\in \mathcal{B}\left(X, Y\right)$. Then the following inequalities hold:

i) $h_{r}\left(K\right) \leq x_{r}\left(K\right) \leq c_{r}\left(K\right) \leq a_{r}\left(K\right)$ and

ii) $h_{r}\left(K\right) \leq y_{r}\left(K\right) \leq d_{r}\left(K\right) \leq a_{r}\left(K\right).$

Lemma 1. ^[5] Let $S, K\in \mathcal{B}\left(X, Y\right),$ then $\left \vert s_{r}\left(K\right) -s_{r}\left(S\right) \right \vert \leq \left \Vert K-S\right \Vert$ for $r = 1, 2, \ldots.$

Let $\omega$ be the space of all real valued sequences. Any vector subspace of $\omega$ is called a sequence space.

In ^[9] the space $Z\left(u, v; l_{p}\right)$ is defined by Malkowsky and Savaş as follows:

where $1 < p < \infty$ and $u = (u_n)$ and $v = (v_n)$ are positive real numbers.

The Cesaro sequence space $ces_{p}$ is defined as (^[10,11,19])

If an operator $K\in \mathcal{B}\left(X, Y\right)$ satisfies $\sum \limits_{n = 1}^{\infty }\left(a_{n}\left(K\right) \right) ^{p} < \infty$ for $0 < p < \infty$, $K$ is defined as an $l_{p}$ type operator in ^[4] by Pietsch. Afterwards $ces$-$p$ type operators which is a new class obtained via Cesaro sequence space is introduced by Constantin ^[12]. Later on Tita in ^[14] proved that the class of $l_{p}$ type operators and $ces$-$p$ type operators coincide.

In ^[15], $\varsigma _{p}^{\left(s\right) },$ the class of $s-$type $Z\left(u, v; l_{p}\right)$ operators is given. For more information about sequence spaces and operator ideals we refer to ^{[1,13,16,18,20]}.

Let $X^{'}$, the dual of $X,$ be the set of continuous linear functionals on $X.$ The map $x^\ast\otimes y:X\rightarrow Y$ is defined by

where $x\in X$, $x^\ast \in$ $X^{'}$ and $y\in Y$.

A subcollection$\ \Im$ of $\mathcal{B}$ is said to be an operator ideal if for each component $\Im \left(X, Y\right) =$ $\Im \cap \mathcal{B} \left(X, Y\right)$ the following conditions hold:

$(i)$ if $x^\ast\in$ $X^{'}$, $y\in Y$, then $x^\ast\otimes y\in \Im \left(X, Y\right),$

$(ii)$ if $L, K\in \Im \left(X, Y\right),$ then $L+K\in \Im \left(X, Y\right),$

$(iii)$ if $L\in \Im \left(X, Y\right),$ $K\in \mathcal{B}\left(X_{0}, X\right)$ and $M\in \mathcal{B}\left(Y, Y_{0}\right),$ then $MLK\in \Im \left(X_{0}, Y_{0}\right)$ ^[6].

Let $\Im$ be an operator ideal and $\rho :\Im \rightarrow$ $\mathbb{R} ^{+}$ be a function on $\Im$. Then, if the following conditions hold:

$(i)$ if $x^\ast\in$ $X^{'}$, $y\in Y$, then $\rho \left(x^\ast\otimes y\right) = \left \Vert x^\ast\right \Vert \left \Vert y\right \Vert;$

$(ii)$ if $\exists C\geq 1$ such that $\rho \left(L+K\right) \leq C\left[\rho \left(L\right) +\rho \left(K\right) \right];$

$(iii)$ if $L\in \Im \left(X, Y\right), K\in \mathcal{B}\left(X_{0}, X\right)$ and $M\in \mathcal{B}\left(Y, Y_{0}\right),$ then $\rho \left(MLK\right) \leq \left \Vert M\right \Vert \rho \left(L\right) \left \Vert K\right \Vert$,

$\rho$ is said to be a quasi-norm on the operator ideal $\Im$ ^[6].

For special case $C = 1$, $\rho$ is a norm on the operator ideal $\Im.$

If $\rho$ is a quasi-norm on an operator ideal $\Im$, it is denoted by $\left[\Im, \rho \right]$. Also if every component $\Im \left(X, Y\right)$ is complete with respect to the quasi-norm $\rho$, $\left[\Im, \rho \right]$ is called a quasi-Banach operator ideal.

Let $\left[\Im, \rho \right]$ be a quasi-normed operator ideal and $\mathcal{J}\in \mathcal{B}\left(Y, Y_0\right)$ be a isometric embedding. If for every operator $K\in \mathcal{B}\left(X, Y\right)$ and $\mathcal{J}K\in \Im \left(X, Y_{0}\right)$ we have $K\in \Im \left(X, Y\right)$ and $\rho \left(\mathcal{J}K\right) = \rho \left(K\right)$, $\left[\Im, \rho \right]$ is called an injective quasi-normed operator ideal. Furthermore, let $\left[\Im, \rho \right]$ be a quasi-normed operator ideal and $\mathcal{S}\in \mathcal{B}\left(X_0, X\right)$ be a quotient map. If for every operator $K\in \mathcal{B}\left(X, Y\right)$ and $K\mathcal{S}\in \Im \left(X_0, Y\right)$ we have $K\in \Im \left(X, Y\right)$ and $\rho \left(K\mathcal{S}\right) = \rho \left(K\right)$, $\left[\Im, \rho \right]$ is called an surjective quasi-normed operator ideal ^[6].

Let $K^{'}$ be the dual of $K$. An $s-$ number sequence is called symmetric (respectively, completely symmetric) if for all $K\in \mathcal{B}$, $s_{r}\left(K\right) \geq s_{r}(K^{'})$ (respectively, $s_{r}\left(K\right) = s_{r}(K^{'})$) ^[6].

Lemma 2. ^[6] The approximation numbers are symmetric, i.e., $a_r(K^{'}) \leq a_r(K)$ for $K\in \mathcal{B}$.

Lemma 3. ^[6] Let $K \in \mathcal{B}$. Then

In addition, if $K$ is a compact operator then $c_r(K^{'}) = d_r(K)$.

Lemma 4. ^[8] Let $K \in \mathcal{B}$. Then

.

The dual of an operator ideal $\Im$ is denoted by $\Im ^{'}$ and it is defined as ^[6]

.

An operator ideal $\Im$ is called symmetric if $\Im \subset \Im ^{'}$ and is called completely symmetric if $\Im = \Im ^{'}$ ^[6].

Let $E = \left(E_{n}\right)$ be a partition of finite subsets of the positive integers which satisfies

for $n \in \mathbb{N^+}.$ In ^[21] Foroutannia defined the sequence space $l_{p}\left(E\right)$ by

with the seminorm $\left \Vert \left \vert \cdot \right \vert \right \Vert _{p, E},$ which defined as:

For example, if $E_{n} = \left \{ 3n-2, 3n-1, 3n\right \}$ for all $n$, then $x = \left(x_{n}\right) \in l_{p}\left(E\right)$ if and only if $\sum \limits_{n = 1}^{\infty }\left \vert x_{3n-2}+x_{3n-1}+x_{3n}\right \vert ^{p} < \infty.$ It is obvious that $\left \Vert \left \vert \cdot \right \vert \right \Vert _{p, E}$ is not a norm, since we have $\left \Vert \left \vert x\right \vert \right \Vert _{p, E} = 0$ while $x = \left(-1, 1, 0, 0, \ldots \right)$ and $E_{n} = \left \{ 3n-2, 3n-1, 3n\right \}$ for all $n$. For the particular case $E_{n} = \left \{ n\right \}$ for $n\in \mathbb{N^+}$ we get $l_{p}\left(E\right) = l_{p}$ and $\left \Vert \left \vert x\right \vert \right \Vert _{p, E} = \left \Vert x\right \Vert _{p}.$

For more information about block sequence spaces, we refer the reader to ^{[17,22,23,24,25]}.

2.   Results

Let $u = \left(u_{n}\right)$ and $v = \left(v_{n}\right)$ be positive real number sequences. In this section, by replacing $l_{p}$ with $l_{p}\left(E\right)$ we get the sequence space $Z\left(u, v; l_{p}\left(E\right) \right)$ defined as follows:

An operator $K\in \mathcal{B}\left(X, Y\right)$ is in the class of s-type $Z\left(u, v; l_{p}\left(E\right) \right)$ if

The class of all s-type $Z\left(u, v; l_{p}\left(E\right) \right)$ operators is denoted by $L_{z, p, E}\left(X, Y\right)$.

In particular case if $E_{n} = \left \{ n\right \}$ for $n = 1, 2, \ldots,$ then the class $L_{z, p, E}\left(X, Y\right)$ reduces to the class $\varsigma _{p}^{\left(s\right) }.$

Conditions used in Theorem 1 hold throughout the remainder of the paper.

Theorem 1. Fix $1 < p < \infty$. If $\sum \limits_{n = 1}^{\infty }\left(u_{n}\right) ^{p} < \infty$ and $\mathcal{M} > 0$ is such that $v_{2k-1}+v_{2k}\leq \mathcal{M}v_{k}, \mathcal{M} > 0$ for all $k \in \mathbb{N}$, then $L_{z, p, E}$ is an operator ideal.

Proof. Let $x^\ast\in X^{'}$ and $y\in Y$. Since the rank of the operator $x^\ast\otimes y$ is one, $s_{n}\left(x^\ast \otimes y\right) = 0$ for $n\geq 2.$ By using this fact

Therefore $x^\ast\otimes y\in L_{z, p, E}(X, Y).$

Let $L, K\in$ $L_{z, p, E}(X, Y).$ Then

To show that $L+K\in L_{z, p, E}(X, Y)$, let us begin with

By using Minkowski inequality we get;

Hence $L+K\in L_{z, p, E}(X, Y).$

Let $M\in \mathcal{B}(Y, Y_{0}),$ $L\in L_{z, p, E}(X, Y)$ and $K\in \mathcal{B}(X_{0}, X).$ Then,

So $MLK\in L_{z, p, E}(X_{0, }Y_{0}).$

Therefore $L_{z, p, E}(X, Y)$ is an operator ideal.

Theorem 2. $\left \Vert K\right \Vert _{z, p, E} = \dfrac{\left(\sum \limits_{n = 1}^{\infty }\left(u_{n}\sum \limits_{k = 1}^{n}\sum \limits_{j\in E_{k}}v_{j}s_{j}\left(K\right) \right) ^{p}\right) ^{\frac{1}{p}}}{\left(\sum \limits_{n = 1}^{\infty }\left(u_{n}\right) ^{p}\right) ^{\frac{1}{p}}v_{1}}$ is a quasi-norm on the operator ideal $L_{z, p, E}.$

Proof. Let $x^\ast\in X^{'}$ and $y\in Y$. Since the rank of the operator $x^\ast\otimes y$ is one, $s_{n}\left(x^\ast\otimes y\right) = 0$ for $n\geq 2.$ Then

Therefore $\left \Vert x^\ast\otimes y\right \Vert _{z, p, E} = \left \Vert x^\ast\right \Vert \left \Vert y\right \Vert.$

Let $L, K\in L_{z, p, E}(X, Y)$. Then

By using Minkowski inequality we get;

Hence

Let $M\in \mathcal{B}(Y, Y_{0}),$ $L\in L_{z, p, E}(X, Y)$ and $K\in \mathcal{B}(X_{0}, X)$

Therefore $\left \Vert K\right \Vert _{z, p, E}$ is a quasi-norm on $L_{z, p, E}.$

Theorem 3. Let $1 < p < \infty.$ $\left[L_{z, p, E}(X, Y), \left \Vert K\right \Vert _{z, p, E}\right]$ is a quasi-Banach operator ideal.

Proof. Let $X, Y$ be any two Banach spaces and $1\leq p < \infty.$ The following inequality holds

for $K\in L_{z, p, E}(X, Y).$

Let $\left(K_{m}\right)$ be Cauchy in $L_{z, p, E}(X, Y).$ Then for every $\varepsilon > 0$ there exists $n_{0}\in \mathbb{N}$ such that

for all $m, l\geq n_{0}.$ It follows that

Then $\left(K_{m}\right)$ is a Cauchy sequence in $\mathcal{B}\left(X, Y \right).$ $\mathcal{B}\left(X, Y\right)$ is a Banach space since $Y$ is a Banach space. Therefore $\left \Vert K_{m}-K\right \Vert \rightarrow 0$ as $m\rightarrow \infty$ for some $K\in \mathcal{B}\left(X, Y\right).$ Now we show that $\left \Vert K_{m}-K\right \Vert _{z, p, E}\rightarrow 0$ as $m\rightarrow \infty$ for $K\in L_{z, p, E}\left(X, Y\right).$

The operators $K_{l}-K_{m},$ $K-K_{m}$ are in the class $\mathcal{B}\left(X, Y\right)$ for $K_{m}, K_{l}, K\in \mathcal{B}\left(X, Y\right).$

Since $K_{l}\rightarrow K$ as $l\rightarrow \infty$ we obtain

It follows from (2.1) that the statement

holds for all $m, l\geq n_{0.}$ We obtain from (2.2) that

Hence we have

Finally we show that $K\in L_{z, p, E}\left(X, Y\right).$

By using Minkowski inequality; since $K_{m}\in L_{z, p, E}\left(X, Y\right)$ for all $m$ and $\left \Vert K_{m}-K\right \Vert _{z, p, E}\rightarrow 0$ as $m\rightarrow \infty,$ we have

which means $K\in L_{z, p, E}\left(X, Y\right).$

Definition 1. Let $\mu = \left(\mu _{i}\left(K\right) \right)$ be one of the sequences $s = \left(s_{n}\left(K\right) \right),$ $c = \left(c_{n}\left(K\right) \right),$ $d = \left(d_{n}\left(K\right) \right),$ $x = \left(x_{n}\left(K\right) \right),$ $y = \left(y_{n}\left(K\right) \right)$ and $h = \left(h_{n}\left(K\right) \right).$ Then the space $L _{z, p, E}^{\left(\mu \right) }$ generated via $\mu = \left(\mu _{i}\left(K\right) \right)$ is defined as

And the corresponding norm $\left \Vert K\right \Vert _{z, p, E}^{\left(\mu \right) }$ for each class is defined as

Proposition 4. The inclusion $L _{z, p, E }^{\left(a\right) }\subseteq L _{z, q, E}^{\left(a\right) }$ holds for $1 < p\leq q < \infty.$

Proof. Since $l_{p}\subseteq l_{q}$ for $1 < p\leq q < \infty$ we have $L _{z, p, E}^{\left(a\right) }\subseteq L _{z, q, E}^{\left(a\right) }.$

Theorem 4. Let $1 < p < \infty.$ The quasi-Banach operator ideal $\left[L _{z, p, E}^{\left(s\right) }, \left \Vert K\right \Vert _{z, p, E}^{\left(s\right) }\right]$ is injective, if the sequence $s_{n}(K)$ is injective.

Proof. Let $1 < p < \infty$ and $K\in \mathcal{B}\left(X, Y\right)$ and $\mathcal{J}\in \mathcal{B}\left(Y, Y_{0}\right)$ be any isometric embedding. Suppose that $\mathcal{J}K\in L _{z, p, E}^{\left(s\right) }\left(X, Y_{0}\right).$ Then

Since $s = \left(s_{n}\right)$ is injective, we have

Hence we get

Thus $K\in L_{z, p, E}^{\left(s\right) }\left(X, Y\right)$ and we have from (2.3)

Hence the operator ideal $\left[L_{z, p, E}^{\left(s\right) }, \left \Vert K\right \Vert _{z, p, E}^{\left(s\right) }\right]$ is injective.

Conclusion 1. [8,p.90-94] Since the number sequences $(c_{n}(K))$ and $(x_{n}(K))$ are injective, the quasi-Banach operator ideals $\left[L_{z, p, E}^{\left(c\right) }, \left \Vert K\right \Vert _{z, p, E}^{\left(c\right) }\right]$ and $\left[L_{z, p, E}^{\left(x\right) }, \left \Vert K\right \Vert _{z, p, E}^{\left(x\right) }\right]$ are injective.

Theorem 5. Let $1 < p < \infty.$ The quasi-Banach operator ideal $\left[L_{z, p, E}^{\left(s\right) }, \left \Vert K\right \Vert _{z, p, E}^{\left(s\right) }\right]$ is surjective, if the sequence ($s_{n}(K)$) is surjective.

Proof. Let $1 < p < \infty$ and $K\in \mathcal{B}\left(X, Y\right)$ and $\mathcal{S}\in \mathcal{B}\left(X_{0}, X\right)$ be any quotient map. Suppose that $K\mathcal{S}\in L_{z, p, E}^{\left(s\right) }\left(X_{0}, Y\right)$. Then

Since $s = \left(s_{n}\right)$ is surjective, we have

Hence we get

Thus $K\in L_{z, p, E}^{\left(s\right) }\left(X, Y\right)$ and we have from (2.4)

Hence the operator ideal $\left[L_{z, p, E}^{\left(s\right) }, \left \Vert K\right \Vert _{z, p, E}^{\left(s\right) }\right]$is surjective.

Conclusion 2. [8,p.95] Since the number sequences $(d_{n}(K))$ and $(y_{n}(K))$ are surjective, the quasi-Banach operator ideals $\left[L_{z, p, E}^{\left(d\right) }, \left \Vert K\right \Vert _{z, p, E}^{\left(d\right) }\right]$ and $\left[L_{z, p, E}^{\left(y\right) }, \left \Vert K\right \Vert _{z, p, E}^{\left(y\right) }\right]$ are surjective.

Theorem 6. Let $1 < p < \infty.$ Then the following inclusion relations holds:

$i$ $L_{z, p, E}^{\left(a\right) }\subseteq L_{z, p, E}^{\left(c\right) }\subseteq L_{z, p, E}^{\left(x\right) }\subseteq L_{z, p, E}^{\left(h\right) }$

ii $L_{z, p, E}^{\left(a\right) }\subseteq L_{z, p, E}^{\left(d\right) }\subseteq L_{z, p, E}^{\left(y\right) }\subseteq L_{z, p, E}^{\left(h\right) }$.

Proof. Let $K\in L_{z, p, E}^{\left(a\right) }$. Then

where $1 < p < \infty.$ And from Proposition 3, we have;

and

So it is shown that the inclusion relations are satisfied.

Theorem 7. For $1 < p < \infty$, $L_{z, p, E}^{\left(a\right) }$ is a symmetric operator ideal and $L_{z, p, E}^{\left(h\right) }$ is a completely symmetric operator ideal.

Proof. Let $1 < p < \infty.$

Firstly, we show that $L_{z, p, E}^{\left(a\right) }$ is symmetric in other words $L_{z, p, E}^{\left(a\right) }\subseteq \left(L_{z, p, E}^{\left(a\right) }\right) ^{'}$ holds. Let $K\in L_{z, p, E}^{\left(a\right) }$. Then

It follows from [6, p.152] $a_{n}\left(K^{'}\right) \leq a_{n}\left(K\right)$ for $K\in \mathcal{B}.$ Hence we get

Therefore $K\in \left(L_{z, p, E}^{\left(a\right) }\right) ^{'}$. Thus $L_{z, p, E}^{\left(a\right) }$ is symmetric.

Now we prove that the equation $L_{z, p, E}^{\left(h\right) } = \left(L_{z, p, E}^{\left(h\right) }\right) ^{'}$ holds. It follows from [8,p.97] that $h_{n}\left(K^{'}\right) = h_{n}\left(K\right)$ for $K\in \mathcal{B}.$ Then we can write

Hence $L_{z, p, E}^{\left(h\right) }$ is completely symmetric.

Theorem 8. Let $1 < p < \infty.$ The equation $L_{z, p, E}^{\left(c\right) } = \left(L_{z, p, E}^{\left(d\right) }\right) ^{'}$ and the inclusion relation $L_{z, p, E}^{\left(d\right) }\subseteq \left(L_{z, p, E}^{\left(c\right) }\right) ^{'}$ holds. Also, for a compact operator $K$, $K\in L_{z, p, E}^{\left(d\right) }$ if and only if $K^{'} \in \left(L_{z, p, E}^{\left(c\right) }\right)$.

Proof. Let $1 < p < \infty.$ For $K\in \mathcal{B}$ it is known from ^[8] that $c_{n}\left(K\right) = d_{n}\left(K^{'}\right)$ and $c_{n}\left(K^{'}\right) \leq d_{n}\left(K\right).$ Also, when $K$ is a compact operator, the equality $c_{n}\left(K^{'}\right) = d_{n}\left(K\right)$ holds. Thus the proof is clear.

Theorem 9. $L_{z, p, E}^{\left(x\right) } = \left(L_{z, p, E}^{\left(y\right) }\right) ^{'}$ and $L_{z, p, E}^{\left(y\right) } = \left(L_{z, p, E}^{\left(x\right) }\right) ^{'}$ hold for $1 < p < \infty.$

Proof. Let $1 < p < \infty.$ For $K\in \mathcal{B}$ we have from ^[8] that $x_{n}\left(K\right) = y_{n}\left(K^{'}\right)$ and $y_{n}\left(K\right) = x_{n}\left(K^{'}\right).$ Thus the proof is clear.

Acknowledgments

The authors would like to thank anonymous referees for their careful corrections and valuable comments on the original version of this paper.

Conflict of interest

The authors declare no conflict of interest.