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Suppose $\mathsf{H}$ is an infinite-dimensional (non-separable preferably) Hilbert space. Consider the space $L(\mathsf{H})$ of all bounded operators on it. Is there $0\neq W\in L(\mathsf{H})$ such that the set

$$\left\{ T\in L(\mathsf{H} )\colon \|W-T\| = \|W+T\| \right\}$$

contains an open ball? Is this set a linear subspace of $L(\mathsf{H})$?

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