We apply numerical radius and spectral radius estimates to the Frobenius companion matrices of monic polynomials to derive new bounds for their zeros and give different proofs of some known bounds.

We present a conjecture which when true would generalize T. Ando's characterization of the numerical radius of (bounded linear) operators on a Hilbert space (see A]). Some evidence for the validity of the conjecture is given. In the nite dimensional case we shall restate the conjecture in terms of convex matrix sets and norms on matrices that are invariant under unitary similarities (u.s.i. nor...

Let m,n ≥ 2 be positive integers. Denote by Mm the set of m×m complex matrices and by w(X) the numerical radius of a square matrix X. Motivated by the study of operations on bipartite systems of quantum states, we show that a linear map φ : Mmn →Mmn satisfies w(φ(A⊗B)) = w(A⊗B) for all A ∈Mm and B ∈Mn if and only if there is a unitary matrix U ∈Mmn and a complex unit ξ such that φ(A⊗B) = ξU(φ1(...

We prove that for a right linear bounded normal operator on a quaternionic Hilbert space (quaternionic bounded normal operator) the norm and the numerical radius are equal. As a consequence of this result we give a new proof of the known fact that a non zero quaternionic compact normal operator has a non zero right eigenvalue. Using this we give a new proof of the spectral theorem for quaternio...

We show that a bounded linear operator A ∈ B(H) is a multiple of a unitary operator if and only if AZ and ZA always have the same numerical radius or the same numerical range for all (rank one) Z ∈ B(H). More generally, for any bounded linear operators A,B ∈ B(H), we show that AZ and ZB always have the same numerical radius (resp., the same numerical range) for all (rank one) Z ∈ B(H) if and on...

We characterize those linear operators on triangular or diagonal matrices preserving the numerical range or radius.

We characterize those linear operators on triangular or diagonal matrices preserving the numerical range or radius.