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.     

Linear Maps Preserving the Closure of Numerical Range on Nest Algebras with Maximal Atomic Nest

Let be a maximal atomic nest on Hilbert space H and denote the associated nest algebra. We prove that a weakly continuous and surjective linear map preserves the closure of numerical range if and only if there exists a unitary operator such that for every or for every , where denotes the transpose of T relative to an arbitrary but fixed base of H. As applications, we get the characterizations of the numerical range or numerical radius preservers on . The surjective linear maps on the diagonal algebras of atomic nest algebras preserving the closure of numerical range or preserving the numerical range (radius) are also characterized. Submitted: January 3, 2001?Revised: December 2, 2001     

A skew normal dilation on the numerical range of an operator

Simple facts about the Poincaré-Neumann double layer potential are used in the construction of a normal dilation, on the numerical range of an arbitrary Hilbert space operator. Recent and old ideas in the theory of the numerical range are unified by this framework. A couple of mapping results for the numerical range are derived.Revised version: 15 July 2002First author partially supported by the National Science Foundation Grant DMS 0100367, Second author supported by STINT, The Swedish Foundation for International Cooperation in Research and Higher Education.     

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.     

On a conjecture of ridge

The conjecture of Ridge on the numerical range of a shift of periodic weights is resolved in the affirmative, i.e., if the weights are nonzero, the numerical range of the corresponding shift is an open disc centered at the origin. The radius of the disc can be expressed as the Perron root of a nonnegative irreducible symmetric matrix. Some related results are obtained.

     

On the Stability Radius of Matrix Polynomials

The stability radius of a matrix polynomial P ( λ) relative to an open region Ωof the complex plane and its relation to the numerical range of P ( λ) are investigated. Using an expression of the stability radius in terms of λon the boundary of Ωand ‖P ( λ) -1 ‖2 , a lower bound is obtained. This bound for the stability radius involves the distances of Ωto the connected components of the numerical range of P ( λ) and can be applied in conjunction with polygonal approximations of the numerical range. The special case of hyperbolic matrix polynomials is also considered.     

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 one-dimensional model for dispersive wave turbulence

Summary A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number. It is established that the quasi-Gaussian random phase hypothesis of weak turbulence theory is an excellent approximation in the numerical statistical steady state. Nevertheless, the predictions of weak turbulence theory fail and yield a much flatter (|k|^−1/3) spectrum compared with the steeper (|k|^−3/4) spectrum observed in the numerical statistical steady state. The reasons for the failure of weak turbulence theory in this context are elucidated here. Finally, an inertial range closure and scaling theory is developed which successfully predicts the inertial range exponents observed in the numerical statistical steady states.     

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.     

Joint numerical range and its generating hypersurface

In this paper, we study the joint numerical range of m-tuples of Hermitian matrices via their generating hypersurfaces. An example is presented which shows the invalidity of an analogous Kippenhahn theorem for the joint numerical range of three Hermitian matrices.     

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.     

Normality and the numerical range

It is well known that if A is an n by n normal matrix, then the numerical range of A is the convex hull of its spectrum. The converse is valid for n ? 4 but not for larger n. In this spirit a characterization of normal matrices is given only in terms of the numerical range. Also, a characterization is given of matrices for which the numerical range coincides with the convex hull of the spectrum. A key observation is that the eigenvectors corresponding to any eigenvalue occuring on the boundary of the numerical range must be orthogonal to eigenvectors corresponding to all other eigenvalues.     

On the Origin and Numerical Range of Bounded Operators

Numerical range has an important applications on spectrum distribution of operators. In this paper, we devoted to characterizing operators whose numerical range contains the origin. Some necessary and sufficient conditions are given by operator decomposition technique and constructive methods. Furthermore, the closeness of the numerical range of a given operator is also investigated.     

Product numerical range in a space with tensor product structure

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.     

On the numerical range in tropical algebra

ABSTRACT

We introduce the concept of numerical range in tropical linear algebra, explore its properties and show in particular that it contains all the tropical eigenvalues. An explicit formula is obtained for the tropical numerical range and an application in the asymptotic analysis of condition number of symmetric matrices is proposed.     

Spektrum,numerischer Wertebereich und IHRE Maximumprinzipien in Banachalgebren

By a function of support a numerical range is defined for elements of complex Banach algebras. The geometric character of the definition allows to proof some results about the numerical range in a very natural way.- In the second part holomorphic perturbations of the numerical range and the spectrum are treated as perturbations of the function of support to get a maximum principle for the numerical range and the convex hull of the spectrum, whereas there is in general no maximum principle for the spectrum. This answers a question raised by A.Brown and R.G.Douglas.

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Ich danke Herrn Prof.Dr.S.Hildebrandt für viele Anregungen. Herrn K.P.Steffen verdanke ich wertvolle Hinweise. Von ihm stammen Satz 2.1 und Beispiel 2.13.     

Spectral inclusion properties of the numerical range in a space with an indefinite metric

In this paper the numerical range of operators (possibly unbounded) in an indefinite inner product space is studied. In particular, we show that the spectrums of bounded positive operators (or the spectrum of unbounded uniformly I-positive operators) are contained in the closure of the I-numerical range.     

The numerical ranges of derivations and quantum physics

A brief discussion of the relevance to Quantum Physics of some familiar concepts in the Theory of Numerical Ranges is given. Some examples of physically relevant problems where the concepts of derivation, numerical range of a derivation, decomposable numerical range and C-numerical range appear naturally, are presented.     

Remarks on the numerical range with respect to a family of projections

In this note we report on the recently introduced concept of the numerical range of a bounded linear operator on a Hilbert space with respect to a family of projections. The importance of this new concept lies in the fact that it unifies and generalizes well-established versions of the numerical range such as the classical numerical range introduced by Toeplitz and Hausdorff, the quadratic numerical range as well as the block numerical range. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation of the Quadratic Numerical Range of Block Operator Matrices

In this paper we establish an approximation of the quadratic numerical range of bounded and unbounded block operator matrices by variational methods. Applications to Hain?CLüst operators are given.