Elliptical higher rank numerical range of some Toeplitz matrices

On Kippenhahn curves and higher-rank numerical ranges of some matrices

“…Formulas (3.4), with some addtional nontrivial computations, provide an alternative approach to the complete description of rank-k numerical ranges of 2-periodic tridiagonal matrices satisfying (3.6). In agreement with [1], they all happen to be elliptical disks. Condition (3.6) holds in particular for tridiagonal Toeplitz matrices.…”

Higher rank numerical ranges of Jordan-like matrices

“…The first case where higher rank numerical ranges of non-normal operators were calculated explicitly is [15], where the author shows that Λ k (T ) is either a disk or empty whenever the n×n matrix T is a power of a shift. In [16] the authors determine the higher rank numerical ranges of direct sums of the form λI ⊕ A 1 ⊕ · · · ⊕ A n , where the matrices A j are 2 × 2, all with the same diagonal; this allows them-via unitary equivalence-to determine the higher numerical ranges of certain 2-Toeplitz tridiagonal matrices. In the cases where the structure of the chain Λ 1 (T ), .…”

“…. , Λ n (T ) is determined explicitly, its structure is fairly simple, going from a fixed type of area (a disk in [15] and an ellipse in [16]) to the empty set. By contrast, the higher rank numerical ranges we find have more variety, see Theorem 3.7.…”

Shell extremal eigenvalues of tridiagonal Toeplitz Matrices

“…an elliptical disc with major length L = 2 1 + cos 4π n+1 (b + c) and foci the eigenvalues λ Theorem 12 and Proposition 13 of[1]). Thus, the convexity of F 2 (T n (c, b, a)) yields that the line segment created by these two eigenvalues, and which contains λ 3 , λ 4 ,• • • , λ n−2 , is contained if F 2 (T n (c, b, a)), which proves item (i).…”