Polynomial numerical hulls of matrices

Field of values of perturbed matrices and quantum states

“…we see that ∥E∥≤ 1 . Since A 2 and (A + E) 2 are Hermitian, by [1,Theorem 4.2], we obtain that V 4 (A) =…”

“…In the following proposition, we list some useful properties of the polynomial numerical hulls of matrices, which will be useful in our discussion. Proposition 3.1 ( [1,2]); Let A ∈ M n and 1 ≤ k ≤ n be a positive integer. Then the following assertions are true: A, A 2 , ..., A k ))} , where conv(.)…”

“…Let ϵ = 1 andA = A 1 ⊕ A 2 ⊕ A 3 , where A 1 = By setting E = E 1 ⊕E 2 ⊕E 3, where E 1 = ∥E∥≤ 1 . Since A 2 and (A + E) 2 are Hermitian, by[1, Theorem 4.2], we obtain thatV 4 (A) = σ(A) = {1, 0, i, √ 3, − √ 3} and V 4 (A+E) = σ(A+E) = {1, − 1 ∈ V 4 (A) + D(0, 1). Thus, V 4 (A) + D(0, 1) ̸ = V 4 1 (A).…”

Joint numerical ranges and compressions of powers of operators

“…For its generalizations see . In general, the structure of can be rather complicated even if is finite‐dimensional, see, for example, .…”