I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try.

Is there an infinite-dimensional Banach space (non-separable preferably) $E$ such that for some non-zero

$T\in L(E)$

the set

$$\{S\in L(E)\colon \|S-T\|=\|S+T\|\}$$

contains an open ball? In fact, I am more interested in the negation:

Is there a Banach space such that for none non-zero $T\in L(E)$ this can happen?

I cannot (dis)prove it even if $E$ is a Hilbert space.

I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try.

Is there an infinite-dimensional Banach space (non-separable preferably) $E$ such that for some $T\in L(E)$ the set

$$\{S\in L(E)\colon \|S-T\|=\|S+T\|\}$$

contains an open ball?

Is there a Banach space such that for none $T\in L(E)$ this can happen? I cannot (dis)prove it even if $E$ is a Hilbert space.

I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try.

Is there an infinite-dimensional Banach space (non-separable preferably) $E$ such that for some $T\in L(E)$ the set

$$\{S\in L(E)\colon \|S-T\|=\|S+T\|\}$$

contains an open ball?

I cannot (dis)prove it even if $E$ is a Hilbert space.