Does there exits any nonseparable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?

$\begingroup$ What is $B(X)$? $\endgroup$ – Taras Banakh Jun 7 '18 at 18:51

$\begingroup$ @TarasBanakh: Edited. $\endgroup$ – Ali Bagheri Jun 7 '18 at 19:03

$\begingroup$ I hope that a counterexample can be found in papers of Koszmider, for example, in impan.pl/~koszmider/badania/fewsur2.pdf or link.springer.com/article/10.5052%2FRACSAM.2010.19 $\endgroup$ – Taras Banakh Jun 7 '18 at 19:19

$\begingroup$ @TarasBanakh, the problem with spaces $K$ such that every operator on $C(K)$ is of the form $gI+W$ where $g\in C(K)$ and $W$ weakly compact, is that there is no clear way how to count weakly compact operators on them. $\endgroup$ – Tomasz Kania Jun 8 '18 at 21:47

$\begingroup$ @TomekKania The number of weakly compact operators is the number of weakly compact sets times the number of continuous operators mapping the unit ball to a given weakly compact set. The number of weakly compact sets should not exceed $X^*^\omega$ and if all weakly compact sets are metrizable (in the weak topology), then the number of continuous operators mapping the unit ball to a given weakly compact set also should not exceed $X^*^\omega$. So, for a Banach space $X$ with ``small'' weakly compact sets, the number of weakly compact operators should not exceed $X^*^\omega$. Right? $\endgroup$ – Taras Banakh Jun 9 '18 at 6:54
An example of a nonseparable Banach space $X$ with $B(X)=\mathfrak c$ is any nonseparable Banach space $X$ whose dual $X^*$ is $w^*$separable and has cardinality $X^*=\mathfrak c$.
This follows from the observation that the map $B(X)\to B(X^*)$, $T\mapsto T^*$, is injective and hence for a countable $w^*$dense set $D$ in $X^*$ we have $$B(X)\le B(X^*)\le (X^*)^{D}=X^*^\omega=\mathfrak c^\omega=\mathfrak c.$$
A ZFCexample of a nonseparable Banach space $X$ whose dual space $X^*$ is $w^*$separable and has cardinality $X^*=\mathfrak c$ is the Banach space $X=C(K)$ of continuous functions on the Alexandroff twoarrow space $K$.
The $w^*$separability of the dual space $X^*$ was proved by Corson, see Theorem 12.43 in this book. The equality $X^*=\mathfrak c$ can be seen analyzing the structure of (probability) measures on the compact space $K$.
Assume Martin's axiom and the negation of CH. Then $2^{\omega_1}=\mathfrak c$. Let $X=\ell_2(\omega_1)$. Every operator on $X$ is determined by its values on a dense set of cardinality $\omega_1$, hence there are at most $$\ell_2(\omega_1)^{\omega_1} = (\omega_1^\omega)^{\omega_1}\leqslant (2^\omega)^{\omega_1}=2^{\omega_1}=\mathfrak c$$ operators on $X$. Consequently, $B(\ell_2(\omega_1))=\mathfrak{c}$.

$\begingroup$ Nice argument. What about under continuum hypothesis? $\endgroup$ – Ali Bagheri Jun 8 '18 at 5:44