We survey results on linear operators leaving invariant different kinds of generalized numerical ranges and numerical radii.

When A is a subalgebra of the commutative Banach algebra C0(X) of all continuous complex-valued functions on a locally compact Hausdorff space X, the spatial numerical range of element of A can be described in terms of positive measures.

Linear dependence of two Hilbert space operators is expressed in terms of equality in modulus of certain sesquilinear and quadratic forms associated with the operators. The forms are based on generalized numerical ranges.

In this paper, we prove that the numerical range of a conjugation on Banach spaces, using connected property, is either unit circle or disc depending dimension given space. When space reflexive, have same result for by applying path-connectedness which applicable to Hilbert setting. addition, show ranges antilinear operators spaces are contained in annuli.

The higher rank numerical ranges of generic matrices are described in terms the components their Kippenhahn curves. Cases tridiagonal (in particular, reciprocal) 2-periodic treated more detail.