The numerical range of a nilpotent operator on a Hilbert space

We prove that the numerical range of an arbitrary nilpotent operator on a complex Hilbert space is a circle (open or closed) with center at and radius not exceeding where is the power of nilpotency of

     

The polynomial numerical hull of a matrix and algorithms for computing the numerical range

Let A be an n × n matrix. In this paper we discuss theoretical properties of the polynomial numerical hull of A of degree one and assemble them into three algorithms to computing the numerical range of A.     

A new concept for block operator matrices:the quadratic numerical range

In this paper a new concept for 2×2-block operator matrices – the quadratic numerical range – is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.     

The block numerical range of matrix polynomials

Block numerical ranges of matrix polynomials, especially the quadratic numerical range, are considered. The main results concern spectral inclusion, boundedness of the block numerical range, an estimate of the resolvent in terms of the quadratic numerical range, geometrical properties of the quadratic numerical range, and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions. As an application, we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials.     

The numerical range of a nonnegative matrix

We offer an almost self-contained development of Perron–Frobenius type results for the numerical range of an (irreducible) nonnegative matrix, rederiving and completing the previous work of Issos, Nylen and Tam, and Tam and Yang on this topic. We solve the open problem of characterizing nonnegative matrices whose numerical ranges are regular convex polygons with center at the origin. Some related results are obtained and some open problems are also posed.     

Another proof of Pell identities by using the determinant of tridiagonal matrix

In this paper, another proof of Pell identities is presented by using the determinant of tridiagonal matrix. It is calculated via the Laplace expansion.     

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.     

Convexity of the upper complex plane part of the numerical range of a quaternionic matrix

The quaternionic numerical range of a matrix with quaternion entries has a convex intersection with the upper half complex plane. The quaternionic analog of the elliptical range theorem is proved.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

Pairs of quaternionic selfadjoint operators with numerical range in a halfplane

The property is studied that two selfadjoint operators on a quaternionic Hilbert space have the joint numerical range in a halfplane bounded by a line passing through the origin. This property is expressed in various ways, in particular, in terms of compressions to two dimensional subpaces, and in terms of linear dependence over the reals. The canonical form for two selfadjoint quaternionic operators in finite dimensional spaces is the main technical tool.     

Numerical ranges of the powers of an operator

The numerical range W(A) of a bounded linear operator A on a Hilbert space is the collection of complex numbers of the form (Av,v) with v ranging over the unit vectors in the Hilbert space. In terms of the location of W(A), inclusion regions are obtained for W(Ak) for positive integers k, and also for negative integers k if A−1 exists. Related inequalities on the numerical radius and the Crawford number are deduced.     

Boltzmann collision operator for the infinite range potential: A limit problem

The conventional Boltzmann collision operator for the infinite range inverse power law model was derived by Maxwell by adopting a collision kernel which is a limit of that for the finite range model by ignoring the glancing angles. Since the interpretation of collision operator for the infinite range potential through limit process to the one with finite range potential is natural in regard to the derivation of the Boltzmann equation. It is the purpose of this paper to clarify the physical meaning of the conventional collision operator for the infinite range inverse power law model through the study of the limiting process of the collision operator as the cutoff radius tends to infinity. We first estimate the extent in which the glancing angles can be ignored in the limiting process. Furthermore we prove that taking limit to collision operator with finite range potential directly will lead to the conventional one with algebraic convergence rate.     

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.     

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

The numerical range and the spectrum of a product of two orthogonal projections

The aim of this paper is to describe the closure of the numerical range of the product of two orthogonal projections in Hilbert space as a closed convex hull of some explicit ellipses parametrized by points in the spectrum. Several improvements (removing the closure of the numerical range of the operator, using a parametrization after its eigenvalues) are possible under additional assumptions. An estimate of the least angular opening of a sector with vertex 1 containing the numerical range of a product of two orthogonal projections onto two subspaces is given in terms of the cosine of the Friedrichs angle. Applications to the rate of convergence in the method of alternating projections and to the uncertainty principle in harmonic analysis are also discussed.     

On the powers and the inverse of a tridiagonal matrix

In this paper, we present an eigendecomposition of a tridiagonal matrix. Tridiagonal matrix powers and inverse are derived. As consequence, we get some relations verified by the coefficients of the inverse and the powers of a tridiagonal matrix.     

On the intrinsic and the spatial numerical range

For a bounded function f from the unit sphere of a closed subspace X of a Banach space Y, we study when the closed convex hull of its spatial numerical range W(f) is equal to its intrinsic numerical range V(f). We show that for every infinite-dimensional Banach space X there is a superspace Y and a bounded linear operator such that . We also show that, up to renormig, for every non-reflexive Banach space Y, one can find a closed subspace X and a bounded linear operator T∈L(X,Y) such that .Finally, we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range, which we call the Bishop-Phelps-Bollobás property, and which is weaker than the uniform smoothness and the finite-dimensionality. We characterize strong subdifferentiability and uniform smoothness in terms of this property.