1 Introduction
There are many partitions of the set of bounded operators on Banach spaces, for example, compact and non-compact operators, and norm-attaining and non-norm-attaining operators. In this article, we consider operators on Banach spaces that in some sense are similar to
$\ell ^p, \, p>1$
, and what weak cluster points of maximizing sequences tells us about the operator. As we will see, these spaces are separable and reflexive, and include Hilbert spaces, the weighted Bergman spaces, and
$\ell ^p$
-spaces. The class of Banach spaces we consider does not include, for example, non-Hilbert
$L^p([0,1])$
and Hardy spaces [Reference Kalton11, Corollary 4.5 and its remark]. It turns out that the infimum of the norm of all weak cluster points of all maximizing sequences to a given operator is either
$0$
or
$1$
. This dichotomy also appears when the supremum is considered instead of the infimum. We, therefore, obtain a partition consisting of three classes of bounded operators, which coincides with the partition obtained from considering norm-attaining operators and operators for which the essential norm coincides with the norm.
The interest of characterizing when the essential norm equals the norm arises not only from Banach space geometry but also from a conjecture concerning the value of the norm of the Hilbert matrix operators on certain weighted Bergman spaces on the complex unit disk,
$A^p_\alpha , \, p>2+\alpha >1$
[Reference Karapetrović13, p. 516]. The essential norm has been determined in [Reference Lindström, Miihkinen and Norrbo15, p. 23] and [Reference Norrbo17, Theorem 1.2]; incremental progress on determining the norm can be found in the introduction in [Reference Norrbo17] (see also [Reference Guanlong, Tian and Wulan7]). Finding a suitable sufficient condition for the norm and essential norm to coincide would be an unconventional way of solving the yet unsolved conjecture.
The Banach algebra of bounded operators and the ideal of compact operators on a Banach space X are denoted
$\mathcal L(X)$
and
$\mathcal K(X)$
, respectively. The essential norm of an operator
$T\in \mathcal L(X)$
is defined as
$\left \Vert T\right \Vert {}_e = \inf _{K\in \mathcal K(X)} \left \Vert T-K\right \Vert $
. Recall that compact operators are always completely continuous, meaning that they map weakly convergent sequences into norm convergent sequences.
Let X be a Banach space. A closed subspace
$Y\subset X$
is said to be an M-ideal if there is a space
$Z\subset X^*$
such that
$X^*=Z\oplus _1 Y^{\perp }$
. A great monograph on M-ideals is [Reference Harmand, Werner and Werner9]. For a given
$p>1$
, a separable Banach space X is said to have property
$(M_p)$
if
$\mathcal K(X\oplus _p X) $
is an M-ideal in
$\mathcal L(X\oplus _p X)$
. This notion was introduced in [Reference Oja and Werner18]. Two useful characterizations of property
$(M_p)$
can be found in [Reference Kalton and Werner12, Corollary 3.6]. A useful property, weaker than
$(M_p)$
, is
$(m_p)$
: For every
$x_n\to 0$
weakly, it holds that
It is worth noting that a space having
$(M_p), \, p>1$
is reflexive (see [Reference Harmand, Werner and Werner9, VI. Proposition 5.2].
We proceed with some further notation. Let
$T\in \mathcal L(X)$
. We say that
$T\in \mathcal L(X)$
is norm attaining if
$\left \Vert T x\right \Vert = \left \Vert T\right \Vert $
for some vector
$x\in B_X$
, and the set of norm-attaining operators is denoted by
$\mathcal {NA} := \mathcal {NA}(X)$
. Whether or not an operator is norm attaining has been examined in, for example, [Reference Dantas, Jung and Martínez-Cervantes4, Reference Pellegrino and Teixeira19]. There is also an open problem concerning the denseness of the norm-attaining operators, initiated in [Reference Bishop and Phelps1] (see also [Reference Lindenstrauss14, Reference Martín Suarez16] for further information). It is easy to see that on a reflexive space, the compact operators are norm attaining, but all norm-attaining operators are not compact.
Let
$\mathcal MS :=\mathcal MS(T)$
be the set of maximizing sequences, that is,
$\mathcal MS := \{(x_n)\subset \partial B_X : \lim _n\left \Vert T x_n\right \Vert = \left \Vert T\right \Vert \}$
. The set of weak cluster points for sequences in
$\mathcal MS$
is given by
$$\begin{align*}\mathcal WC := \mathcal WC(T) :=\bigcup_{(x_n)\in \mathcal MS}\bigcap_k \overline{(x_n)_{n\geq k}}^w, \end{align*}$$
which is nonempty if the space X is reflexive. For a separable reflexive Banach space, we proceed to define the minimal and maximal weak maximizing limit to be
$$\begin{align*}\mathcal ML_{\min} := \mathcal ML_{\min}( T) := \inf \{\left\Vert x\right\Vert : x\in\mathcal WC(T) \} \end{align*}$$
and
$$\begin{align*}\mathcal ML_{\max} := \mathcal ML_{\max}( T) := \sup \{\left\Vert x\right\Vert : x\in\mathcal WC(T) \}, \end{align*}$$
respectively. Notice that
$ \mathcal ML_{\min } , \mathcal ML_{\max } \in [0,1] $
, and by a diagonalization argument the infimum and supremum are attained. Finally, we denote the unit norm weakly null sequences by
$W_0$
.
We proceed by presenting the main result, Theorem 2.1, followed by some examples in Section 2. Section 3 contains some alternative proofs of the parts of the main result that used the essential norm formula stated in [Reference Werner21] (without proof). We also provide a short proof of the essential norm formula using Theorem 2.1.
2 The main result and examples
The following theorem also states that if
$ \mathcal WC\setminus (\{0\}\cup \partial B_X)$
is not empty, then
$\mathcal ML_{\min }=0$
and
$\mathcal ML_{\max }=1$
, creating the dichotomy
$\mathcal ML_{\min },\mathcal ML_{\max }\in \{0,1\}$
.
Theorem 2.1 Let X be a separable Banach space with
$(M_p)$
and let
$T\in \mathcal L(X)$
. It holds that
-
•
$\mathcal ML_{\min }<1$
if and only if
$\mathcal ML_{\min } = 0$
if and only if
$\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e$
; -
•
$\mathcal ML_{\max }>0$
if and only if
$\mathcal ML_{\max } = 1$
if and only if
$T\in \mathcal {NA}$
.
Remark 2.2 From Theorem 2.1, it follows that
$\mathcal L(X)$
can be partitioned into three classes:
-
•
$\mathcal ML_{\min }=\mathcal ML_{\max }=0$
if and only if
$\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e$
and
$T\notin \mathcal {NA}$
, -
•
$\mathcal ML_{\min } = 0$
and
$\mathcal ML_{\max } = 1$
if and only if
$\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e$
and
$T\in \mathcal {NA}$
, and -
•
$\mathcal ML_{\min }=\mathcal ML_{\max }=1$
if and only if
$\left \Vert T\right \Vert> \left \Vert T\right \Vert {}_e$
and
$T\in \mathcal {NA}$
.
The implication
$T\notin \mathcal {NA} \ \Rightarrow \ \left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e$
also follows from [Reference Werner21, Proposition 8(i)].
Remark 2.3 It is clear that
$$\begin{align*}T\in \mathcal{NA} \ \Longrightarrow \ \mathcal ML_{\max} = 1 \ \Longrightarrow \ \mathcal ML_{\max}>0. \end{align*}$$
The implication
$\mathcal ML_{\max }>0 \Rightarrow T\in \mathcal {NA} $
for all
$T\in \mathcal L(X)$
is the definition of X having the weak maximizing property, which has been proved for the spaces
$\ell ^p, \, p>1$
in [Reference Pellegrino and Teixeira19]. The proof for reflexive spaces with
$(m_p)$
follows from the calculations done on the 11 last lines of the proof of Theorem 1 in [Reference Pellegrino and Teixeira19]. Moreover, this proof shows that if
${(x_n)\in \mathcal MS}$
and
$x_n\to x_0$
weakly for some
$x_0\in B_X\setminus \{0\}$
, then T attains its norm at
$x_0/\left \Vert x_0\right \Vert $
.
Proof of Theorem 2.1
The proof of the equivalences involving
$\mathcal ML_{\max }$
and
$\mathcal {NA}$
are contained in Remark 2.3. Moreover, it is clear that
$ \mathcal ML_{\min } = 0 $
implies both
${\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e} $
and
$ \mathcal ML_{\min } <1 $
. In [Reference Werner21] on page 499, after the proof of Lemma 5, it is claimed (without proof) that
$$ \begin{align} \left\Vert T\right\Vert{}_e = \sup_{(x_n)\in W_0} \limsup_n \left\Vert T x_n \right\Vert \end{align} $$
holds if X has
$(M_p)$
. It follows that
$\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e \ \Rightarrow \ \mathcal ML_{\min } = 0 $
. Finally, assume
${T\in \mathcal L(X)}$
is an operator with
$\mathcal ML_{\min } <1 $
. If
$\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e$
, then
$\mathcal ML_{\min } = 0 $
by (2.1). If
${\left \Vert T\right \Vert> \left \Vert T\right \Vert {}_e}$
, the same conclusion,
$\mathcal ML_{\min } = 0 $
, follows from [Reference Siju20, Lemma 3.4].
We begin by providing some examples of spaces satisfying
$(M_p)$
for some
$p>1$
.
Example 2.4
-
• The fact that Hilbert spaces satisfy
$(M_2)$
was proved in [Reference Dixmier5, Theorem 3]. -
• In [Reference Hennefeld10, Corollary 2.2] it is proved that
$ \mathcal K(\ell ^p) $
is an M-ideal in
$ \mathcal L(\ell ^p) $
, for
$ p>1$
. Since
$\ell ^p \oplus _p \ell ^p$
is isometrically isomorphic to
$\ell ^p$
, it follows that
$\ell ^p, \, p>1$
satisfies
$(M_p)$
. -
• The weighted Bergman spaces
$A^p_v, \, p\geq 1$
consists of complex-valued analytic functions on the open, complex, unit disk,
$f\colon \mathbb {D} \to \mathbb {C}$
, and are defined as where
$$\begin{align*}A^p_v := \bigg\{ f\in \mathcal H(\mathbb{D}) : \left\Vert f\right\Vert := \bigg( \int _{\mathbb{D}} \left\vert f(z)\right\vert{}^p v(z) \, dA(z) \bigg)^{\frac{1}{p}}<\infty \bigg\}, \end{align*}$$
$dA(z)$
is the normalized uniform area measure
$dA(x+iy) = dx \, dy/\pi $
and
$L^1(\mathbb {D},dA)\ni v\colon \mathbb {D} \to ]0,\infty [$
is a suitable (weight) function, for example, continuous, radial (
$v(z) = v(\left \vert z\right \vert )$
), and monotone function. Using [Reference Kalton and Werner12, Corollary 3.6], one can prove that the reflexive weighted Bergman spaces
$A^p_v, \, p>1$
has
$(M_p)$
(see also [Reference Kalton and Werner12, Corollary 4.8]). The fact that
$A^p_v$
has
$(m_p)$
can be proved similarly to [Reference Brezis and Lieb2, Proof of Theorem 2]; some details are provided in the corrigendum to [Reference Lindström, Miihkinen and Norrbo15].
Next, we give examples of operators in each of the three classes presented in Remark 2.2.
Example 2.5 Consider the weighted Bergman spaces
$A^p_\alpha , \, p>1,\alpha >-1$
, where the weight function is given by
$v(z) = (1+\alpha )(1-\left \vert z\right \vert {}^2)^\alpha $
. For general weights, the only additional assumption we impose in this example, other than a weight yielding the
$(M_p)$
property, is that the evaluation maps
$\delta _z\colon A^p_v \to \mathbb C \colon f\mapsto f(z), \, z\in \mathbb {D}$
are bounded. A good monograph on the spaces
$A^p_\alpha $
is [Reference Zhu22]. In [Reference Zhu22, Theorem 4.14], it is proved that the evaluation maps are bounded on
$A^p_\alpha $
.
Consider the multiplication operators
$M_g \colon f \mapsto gf$
on
$A^p_\alpha $
, where
$g\colon \mathbb {D} \to \mathbb {C}$
is bounded and analytic. If g is constant, then
$M_g = c I$
for some
$c\in \mathbb {C}$
. Clearly,
$M_g$
attains its norm at every unit vector in
$A^p_\alpha $
proving
$\mathcal ML_{\max }(M_g) = 1$
. For
$n=0,1,2\ldots $
, let
$e_n$
denote the normalized version of the basis vector
$z^n$
. Then,
$(e_n)$
is a weakly null sequence, proving
$\mathcal ML_{\min }(M_g) = 0$
. Now, assume that g is not constant. It is well known that
$\left \Vert M_g\right \Vert = \sup _{z\in \mathbb {D}} \left \vert g(z)\right \vert $
; the lower bound for the norm can be obtained using the bounded evaluation maps
$\delta _z$
. Moreover, partitioning the open unit disk into a closed disk with radius
$1/2$
,
$\overline {(1/2)\mathbb {D}}$
, and its complement, it follows from the maximum modulus principle and the fact that any analytic function has at most finitely many zeros in
$\overline {(1/2)\mathbb {D}}$
that
$\left \Vert M_g f\right \Vert < \sup _{z\in \mathbb {D}} \left \vert g(z)\right \vert $
for all
$f\in B_{A^p_\alpha }$
. Hence,
$M_g$
is not norm attaining.
To summarize, for any bounded analytic function
$g\colon \mathbb {D} \to \mathbb {C}$
, we have
$M_g\in \mathcal L(A^p)$
and
$\mathcal ML_{\min }(M_g) = 0$
. If g is constant, then
$\mathcal ML_{\max }(M_g) = 1$
, else
$\mathcal ML_{\max }(M_g) = 0$
.
Example 2.6 The null operator satisfies
$\mathcal ML_{\min } = 0, \mathcal ML_{\max } = 1$
, but other compact operators K cannot have
$\mathcal ML_{\min }(K) = 0$
, therefore,
$\mathcal ML_{\min }(K) =\mathcal ML_{\max }(K) = 1$
. This can be seen, for example, by the fact that they are completely continuous, or the fact that
$\left \Vert K\right \Vert>0=\left \Vert K\right \Vert {}_e$
in conjunction with Theorem 2.1.
Example 2.7 On the complex-valued (or real valued)
$\ell ^p(\mathbb N)$
-space, the pointwise multiplication operator can belong to all three classes, depending on the symbol
$b\in \ell ^\infty $
. The norm is given by
$\left \Vert M_b\right \Vert = \sup _n \left \vert b(n)\right \vert $
and the essential norm is given by
$\left \Vert M_b\right \Vert {}_e = \limsup _n \left \vert b(n)\right \vert $
. We can see that the maximum modulus principle for analytic functions is the reason
$\left \Vert M_g\right \Vert {}_e < \left \Vert M_g\right \Vert $
never happens on Bergman spaces. It is, however, easy to see that both
$\left \Vert M_b\right \Vert {}_e < \left \Vert M_b\right \Vert $
and
$\left \Vert M_b\right \Vert {}_e = \left \Vert M_b\right \Vert $
can happen on
$\ell ^p$
. Moreover, if
$\left \vert b(k)\right \vert <\sup _n \left \vert b(n)\right \vert $
for all k, then
$\mathcal ML_{\max } = 0$
.
3 Some alternative proofs
In the proof of Theorem 2.1, the two statements
$\mathcal ML_{\min }<1$
implies
$\mathcal ML_{\min } = 0$
, and
$\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e$
implies
$\mathcal ML_{\min } = 0 $
relied on the essential norm formula (2.1). We proceed with proving these statements without the use of (2.1), which in turn would prove Theorem 2.1 without the use of (2.1). We end the section with a simple proof of the formula for the essential norm (2.1) using Theorem 2.1.
Proposition 3.1 Let X be a separable reflexive Banach space with
$(m_p)$
, and let
${T\in \mathcal L(X)}$
. Then
$\mathcal ML_{\min }<1$
implies
$\mathcal ML_{\min } = 0$
.
Proposition 3.1 follows immediately from the following lemma, whose proof share some similarities with [Reference Siju20, Proof of Lemma 3.4].
Lemma 3.2 Let
$ X$
be a separable reflexive Banach space that satisfies property
$(m_p)$
. Let
$T\in \mathcal L(X)$
and
$x_0\in \mathcal WC(T)$
. There exists a sequence
$(z_n)\in W_0$
such that
$$\begin{align*}\left\Vert T\right\Vert{}^p (1 - \left\Vert x_0\right\Vert{}^p) = \lim_n \left\Vert Tz_n\right\Vert{}^p (1-\left\Vert x_0\right\Vert{}^p ). \end{align*}$$
Proof Let
$(y_n)\in \mathcal MS$
with
$y_n\to x_0$
weakly. By going to a subsequence if necessary, we can assume that
$ \lim _n \left \Vert y_n - x_0 \right \Vert {}^p$
exists, and that either
$ \lim _n \left \Vert y_n - x_0\right \Vert = 0$
or
$ \inf _n\left \Vert y_n - x_0\right \Vert>0$
. If
$ \lim _n \left \Vert y_n - x_0\right \Vert = 0$
, we have
$\left \Vert x_0\right \Vert =1$
and we are done. If
$\inf _n\left \Vert y_n - x_0\right \Vert>0$
, put
$z_n = (y_n-x_0)/\left \Vert y_n-x_0\right \Vert $
. Property
$(m_p)$
yields
$ 1 - \left \Vert x_0 \right \Vert {}^p = \limsup _n \left \Vert y_n - x_0 \right \Vert {}^p $
. Similarly, we have
$ \left \Vert T \right \Vert {}^p - \left \Vert Tx_0 \right \Vert {}^p = \limsup _n \left \Vert T(y_n - x_0) \right \Vert {}^p $
. It follows that
$\left \Vert T \right \Vert {}^p - \left \Vert Tx_0 \right \Vert {}^p = \limsup _n \left \Vert Tz_n \right \Vert {}^p (1-\left \Vert x_0 \right \Vert {}^p) $
. Combining this with the elementary inequalities
$$\begin{align*}\left\Vert T \right\Vert{}^p - \left\Vert Tx_0 \right\Vert{}^p \geq \left\Vert T \right\Vert{}^p (1-\left\Vert x_0\right\Vert{}^p) \geq \limsup_n \left\Vert Tz_n \right\Vert{}^p (1-\left\Vert x_0 \right\Vert{}^p) \end{align*}$$
yields the statement.
Property
$(m_p)$
yields that
$\mathcal ML_{\min }<1$
is equivalent to having a maximizing sequence with no norm-convergent subsequences. Utilizing the fact that
$\mathcal ML_{\min } = 0 \ \Rightarrow \ \left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e$
, Proposition 3.1 can be expressed as follows.
Corollary 3.3 Let X be a separable reflexive Banach space with
$(m_p)$
, and let
${T\in \mathcal L(X)}$
. Then
$\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e$
if there is a maximizing sequence for T with no norm-convergent subsequences.
For the next proof, we define
$\mathcal MV := \mathcal MV(T)$
to be the set of vectors in
$x\in B_X$
satisfying
$\left \Vert T x\right \Vert = \left \Vert T\right \Vert $
. This set is closed, and by definition non-empty if and only if
$T $
is norm attaining. Moreover, if X has
$(M_p)$
, then
$\mathcal K(X)$
is an M-ideal in
$\mathcal L(X)$
. This follows immediately from comparing [Reference Kalton and Werner12, Corollary 3.6] with [Reference Kalton and Werner12, Theorem 2.13].
Proposition 3.4 Let X be a separable Banach space with
$(M_p)$
and let
$T\in \mathcal L(X)$
. It holds that
$\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e \ \Rightarrow \ \mathcal ML_{\min } = 0 $
.
Proof We use contraposition to prove this statement. It is sufficient to assume
$T\not \equiv 0$
and
$\mathcal ML_{\min }>0$
. By Proposition 3.1, we may assume
$\mathcal ML_{\min } = 1$
. In order to prove that
$\left \Vert T\right \Vert> \left \Vert T\right \Vert {}_e$
, let
$(K_n)\subset B_{\mathcal K(X)}$
be a sequence satisfying
$ \lim _n \left \Vert I-K_n\right \Vert = 1$
and
$ \lim _n \left \Vert (I - K_n) x\right \Vert = 0$
for every
$x\in X$
; this is possible due to [Reference Kalton11, Theorem 2.4(4)] (see also [Reference Harmand and Lima8, Lemma 5.1]). Moreover, property
$(m_p)$
implies that any sequence
$(x_n)\subset B_X$
that converges weakly to an
$x_0\in \partial B_X$
must satisfy
$ \lim _n \left \Vert x_n-x_0\right \Vert = 0$
. Hence, since the space is reflexive and
$\mathcal ML_{\min } = 1$
, every
$(x_n)\in \mathcal MS$
will have a norm convergent subsequence. It follows that
$MV$
is compact and
$$ \begin{align} \overline{\{x_n\}}^{\left\Vert\cdot\right\Vert} \cap \mathcal MV \neq \emptyset \quad \text{ for every } (x_n)\in \mathcal MS. \end{align} $$
Since
$\mathcal MV$
is compact, we have
$$\begin{align*}\lim_n\sup_{x\in\mathcal MV} \left\Vert(I-K_n)Tx\right\Vert = 0. \end{align*}$$
It follows that there is an open set
$N\supset \mathcal MV$
that is independent of n such that
$$ \begin{align} \limsup_n \sup_{x\in N} \left\Vert(I-K_n)Tx\right\Vert <\left\Vert T\right\Vert. \end{align} $$
Furthermore, the closed set
$B_X\setminus N$
is disjoint from
$\mathcal MV$
, therefore, (3.1) yields
$$\begin{align*}(B_X\setminus N)^{\mathbb N} \cap \mathcal MS = \emptyset. \end{align*}$$
We can now conclude that
$$\begin{align*}\limsup_n \sup_{x\in B_X \setminus N} \left\Vert(I-K_n)Tx\right\Vert \leq \limsup_n \left\Vert I-K_n\right\Vert \sup_{x\in B_X \setminus N} \left\Vert Tx\right\Vert <\left\Vert T\right\Vert, \end{align*}$$
and hence, using (3.2),
$$\begin{align*}\left\Vert T\right\Vert{}_e = \inf_K \max\Big\{ \sup_{x\in N\cap B_X }\left\Vert(T-K)x\right\Vert , \sup_{x\in B_X\setminus N }\left\Vert(T-K)x\right\Vert \Big\} < \left\Vert T\right\Vert.\\[-42pt] \end{align*}$$
We end the section with a proof of the essential norm formula for operators on separable Banach spaces with
$(M_p)$
, stated on [Reference Werner21, p. 499].
Proposition 3.5 Let X be a separable Banach space with
$(M_p)$
. The essential norm of an operator
$T\in \mathcal L(X)$
can be represented as
$$\begin{align*}\left\Vert T\right\Vert{}_e = \sup_{(x_n)\in W_0} \limsup_n \left\Vert Tx_n\right\Vert. \end{align*}$$
Proof Since the space of compact operators
$\mathcal K(X)$
is proximal in
$\mathcal L(X)$
[Reference Harmand, Werner and Werner9, II. Proposition 1.1], there is a
$K\in \mathcal K(X)$
such that
$$\begin{align*}\left\Vert T\right\Vert{}_e =\left\Vert T-K\right\Vert. \end{align*}$$
Now, we can apply Theorem 2.1 to the operator
$T-K$
, to obtain that either
$\mathcal ML_{\min }(T-K) = 0$
, in which case we are done, or
$\left \Vert T-K\right \Vert> \left \Vert T-K\right \Vert {}_e$
. However, the latter cannot occur since K is the compact operator minimizing
$\left \Vert T-K\right \Vert $
, so
$\left \Vert T-K\right \Vert = \left \Vert T-K\right \Vert {}_e$
and we are done.
Inspired by a comment of Rishit R Rajpopat concerning a preprint of this work.
Open question: Given a Banach space X, characterize operators
$T\in \mathcal {NA}$
for which
$\left \Vert T\right \Vert> \left \Vert T\right\Vert{}_e$
and
$\operatorname {\mathrm {span}} \mathcal MV$
is infinite dimensional, if such exists. The question was initially posed in the setting of X satisfying
$M_p$
in my communication with R. R Rajpopat.
4 Some useful remarks concerning the hierarchy of assumptions
For a given
$\lambda \geq 1$
, a Banach space is said to have the
$\lambda $
-compact approximation property if for every
$\epsilon>0$
and compact set S, there is a compact operator K with
${\left \Vert K\right \Vert \leq \lambda }$
such that
$\sup _{x\in S}\left \Vert (I-K)x\right \Vert <\epsilon $
. If we don’t demand a uniform bound on the norm of the compact operators, we say the space has the compact approximation property, and if
$\lambda =1$
, it has the metric compact approximation property. For a separable reflexive space, all of these approximation properties coincide, see [Reference Cho and Johnson3, Proposition 1] (and also [Reference Godefroy and Saphar6, Remark 1.4]).
It was mentioned in the introduction that a space is reflexive if it has
$(M_p), \, p>1$
. As a consequence, [Reference Kalton and Werner12, Corollary 3.6] implies a separable Banach space has
$(M_p)$
if and only if it is reflexive, has
$(m_p)$
and the compact approximation property.
We end the section with the following remark.
Remark 4.1 Concerning the assumptions in Theorem 2.1, it is sufficient to assume X is a separable reflexive Banach space with
$(m_p)$
, except for the implication: If
${\left \Vert T\right \Vert = \left \Vert T\right \Vert {}_e }$
, then there is a weakly null maximizing sequence, see the proof of Proposition 3.4. For this implication, we need X to have the compact approximation property.
Acknowledgements
The author would like to thank Sreejith Siju for mentioning Lemma 3.4 in [Reference Siju20] and Rishit R Rajpopat for questioning an earlier version of the proof of Proposition 3.4, which helped to improve the proof.
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