Essential numerical range and $C$-numerical range for unbounded operators

. We introduce two new concepts for unbounded operators T in a Hilbert space, the essential numerical range W e 5 ( T ) of type 5 and the C -numerical range W C ( T ) . Our first main result clarifies the relation of W e 5 ( T ) to the essential numerical range W e ( T ) , answering an open problem of Bögli, Marletta and Tretter’s (2020) by employing the Bessaga–Pełczyński selection theorem from Banach space theory. It turns out that W e 5 ( T ) ⊂ W e ( T ) and we establish sharp conditions for equality. An example for strict inclusion shows that W e ( T ) may be a half-plane, while W e 5 ( T ) only a line. We also show that W e 5 ( T ) is convex and that it contains the convex hull of the essential spectrum. Our second main result reveals a geometric relation between W e 5 ( T ) and W C ( T ) . We show that, for finite-rank operators C , W C ( T ) is star-shaped with star-centre (Tr C ) W e 5 ( T ) , generalizing a result for bounded operators where W e 5 ( T ) = W e ( T ) .