On symmetric points with respect to the numerical radius norm

We study left symmetric and right symmetric points with respect to the numerical radius orthogonality (respectively, known as nr-left symmetric operators and nr-right symmetric operators) in the setting of both Hilbert spaces and Banach spaces. We prove that a bounded linear operator T on a complex Hilbert space is nr-left symmetric if and only if T is the zero operator, provided that T attains its numerical radius. We also prove that a nonzero compact normal operator on an infinite-dimensional complex Hilbert space cannot be nr-right symmetric. We then study nr-left symmetry and nr-right symmetry in the setting of Banach spaces and obtain separate necessary and sufficient conditions for the same. Next, we obtain complete characterizations of nr-left and nr-right symmetric operators on some particular Banach spaces.