The spectrum of an operator in a Hilbert space is an important topic in functional analysis, with many applications in both theoretical and applied contexts. Alongside the spectrum, the numerical range—which is the image of the unit sphere under a certain quadratic form—is a useful but less widely known tool. The geometric structure of the numerical range depend strongly on the ground field being the complex numbers or the skew-field of Hamilton's quaternions. In the complex case, the numerical range of both bounded and unbounded operators has been deeply studied and it can be used to help to locate the spectrum and give important information about operator's behavior. However, when extending this concept to quaternionic Hilbert spaces, the non-commutative nature of quaternions introduces substantial challenges. Classical tools from complex operator theory no longer apply directly and must be carefully reformulated. While recent work has begun to address the numerical range for bounded operators in quaternionic Hilbert spaces, the unbounded case remains mostly unexplored. In this talk, I explore the properties and shape of the numerical range of bounded and unbounded operators on quaternionic Hilbert spaces. I discuss its relation with the spectrum, how it reflects important features of the operator, and what new ideas are needed to work in this non-commutative setting. This is a joint work with Luís Carvalho and Sérgio Mendes.

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