Some Generalized Numerical Ranges of Real-linear Operators on Hilbert Spaces
Arup Majumdar, Vladimir Muller
Abstract
This paper introduces and systematically investigates several classes of numerical ranges associated with real-linear operators on Hilbert spaces, including the essential numerical range, maximal numerical range, essential maximal numerical range, q-numerical range, and trace numerical range. We establish fundamental structural properties of these sets and characterize their geometric shape. In particular, we analyze how the real-linear nature of the operators influences the geometry and spectral behaviour of the corresponding numerical ranges, thereby extending and enriching the theory of numerical ranges from complex-linear to real-linear operator settings.
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