The spectrum and the numerical range of a bounded operator is arguably two of the most important concepts associated with linear operators. Their properties are greatly influenced by the underlying field. For instance, the usual definition of spectrum cannot be used in the setting of infinite dimensional quaternionic Hilbert spaces, leading to the notion of S-spectrum. Regarding the numerical range, in complex Hilbert spaces, the Toeplitz-Hausdorff Theorem ensures that the numerical range is always convex. However, in the setting of quaternionic Hilbert spaces, this convexity property may no longer hold. Moreover, it is generally challenging to describe the numerical range for operators in quaternionic spaces, and its shape remains largely unpredictable. A notable exception occurs for normal operators, where more is known about both convexity and the structure of the numerical range. In this presentation, we focus on numerical range of bounded linear operators in quaternionic Hilbert spaces and explore its connection to the S-spectrum of these operators. Our goal is to extend several classical results from complex Hilbert space theory to the quaternionic case. In particular, we provide a detailed characterization of the numerical range for normal operators in quaternionic Hilbert spaces. This is joint work with Luís Carvalho, Cristina Diogo and Helena Soares.

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Numerical range,quaternionic Hilbert Space,S-spectrum