Lucas' theorem and numerical range

The Lucas' theorem is generalized to the numerical ranges of some 3?×?3 companion matrices. We determine monic polynomials of degree 3 which assert the generalization. Examples are provided to show the generalization is restricted which gives a negative answer to the question raised by Zemanek [J. Zemanek, The derivative and linear algebra, Kolo Mlodych Matematykow, Ogolnopolskie Warsztaty, dla Mlodych Matematykow, Krakow, 2003, pp. 207–212.].     

Birkhoff's theorem and multidimensional numerical range

We show that, under certain conditions, Birkhoff's theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections. Using this result, we study the relation between the spectrum of a self-adjoint operator A and its multidimensional numerical range. It turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator A. Every collection of eigenvalues which can be obtained by the Rayleigh-Ritz formula generates an extreme point of the multidimensional numerical range. However, it may also have other extreme points.     

A short proof of a theorem on the numerical range of a normal quaternionic matrix

A short proof of a recent result obtained by So, Thompson and Zhang is given.     

A remark on numerical range of semi-hyponormal operators

Let U be the unilateral shift on ?². For any complex numbers α and β, put T = αU + βU* and S = T ². Then we show that the operator S is convexoid.     

A symmetric numerical range for matrices

Summary For each normv on ⁿ, we define a numerical rangeZ _v, which is symmetric in the sense thatZ _v=Z_vD, wherev ^D is the dual norm.We prove that, fora ⁿⁿ,Z _v(a) contains the classical field of valuesV(a). In the special case thatv is thel ₁-norm,Z _v(a) is contained in a setG(a) of Gershgorin type defined by C. R. Johnson.Whena is in the complex linear span of both the Hermitians and thev-Hermitians, thenZ _v(a),V(a) and the convex hull of the usualv-numerical rangeV _v(a) all coincide. We prove some results concerning points ofV(a) which are extreme points ofZ _v(a).Part of this research was done while the authors were at the Mathematische Institut, Technische Universität, München, West Germany. The first author presented these results at the Seminar on Matrix Theory (Positivity and Norms) held in Munich in December, 1974. The second author also acknowledges support from the National Science Foundation under grant GP 37978X.     

A simple proof of the elliptical range theorem

A short proof is given of the elliptical range theorem concerning the numerical range of a complex matrix.

     

Alternative proofs of a theorem of Moyls and Marcus on the numerical range of a square matrix

Moyls and Marcus [4] showed that for n≤4,n×n an complex matrix A is normal if and only if the numerical range of A is the convex hull of the eigenvalues of A. When n≥5, there exist matrices which are not normal, but such that the numerical range is still the convex hull of the eigenvalues. Two alternative proofs of this fact are given. One proof uses the known structure of the numerical range of a 2×2 matrix. The other relies on a theorem of Motzkin and Taussky stating that a pair of Hermitian matrices with property L must commute.     

A functional calculus based on the numerical range: applications

We develop a functional calculus for both bounded and unbounded operators in Hilbert spaces based on a simple inequality related to polynomial functions of a square matrix and involving the numerical range. We present some applications in different areas of mathematics.     

A generalized numerical range: the range of a constrained sesquilinear form

Let M_n denote the algebra of all nxn complex matrices. For a given q?C with ∣Q∣≤1, we define and denote the q-numerical range of A?M_n by

W_q (A)={x ^? Ay:x,y?C ⁿ , _{x ^? x?y ^? y=1,x ^? y=q} }

The q-numerical radius is then given by r_q (A)=sup{∣z∣:z?W _q (A)}. When q=1,W _q (A) and r _q (A) reduce to the classical numerical range of A and the classical numerical radius of A, respectively. when q≠0, another interesting quantity associated with W _q (A) is the inner q-numerical radius defined by [rtilde] _q (A)=inf{∣z∣:z?W _q (A)}

In this paper, we describe some basic properties of W _q (A), extending known results on the classical numerical range. We also study the properties of r_q considered as a norm (seminorm if q=0) on M_n .Finally, we characterize those linear operators L on M_n that leave W_q ,r_q of [rtilde]_q invariant. Extension of some of our results to the infinite dimensional case is discussed, and open problems are mentioned.