Positive sesquilinear form measures and generalized eigenvalue expansions

Positive operator measures (with values in the space of bounded operators on a Hilbert space) and their generalizations, mainly positive sesquilinear form measures, are considered with the aim of providing a framework for their generalized eigenvalue type expansions. Though there are formal similarities with earlier approaches to special cases of the problem, the paper differs e.g. from standard rigged Hilbert space constructions and avoids the introduction of nuclear spaces. The techniques are predominantly measure theoretic and hence the Hilbert spaces involved are separable. The results range from a Naimark type dilation result to direct integral representations and a fairly concrete generalized eigenvalue expansion for unbounded normal operators.     

The convexity of the range of three Hermitian forms and of the numerical range of sesquilinear forms

This note contains a demonstration of the convexity of the joint range of three Hermitian forms, and, as an immediate corollary, of the convexity of the numerical range of arbitrary sesquilinear forms.     

The constrained bilinear form and the C-numerical range

Let V be an n-dimentional unitary space with inner product (·,·) and S the set {x∈V:(x, x)=1}. For any A∈Hom(V, V) and $\text{q∈}\text{C}$ with ∣q∣?1, we define

$\text{W(A:q)={(Ax, y):x, y∈S, (x, y)=q}}$

. If q=1, then W(A:q) is just the classical numerical range {(Ax, x):x∈S}, the convexity of which is well known. Another generalization of the numerical range is the C-numerical range, which is defined to be the set

$\text{W}{}_{\mathrm{C}}\text{(A)={}\text{tr}\text{(CU}{}^1\text{AU):U}\text{unitary}\text{}}$

where C∈Hom(V, V). In this note, we prove that W(A:q) is always convex and that W_C(A) is convex for all A if rank C=1 or n=2.     

Conditions for the generalized numerical range to be real

Necessary and sufficient conditions are given for the C-numerical range of a matrix A to be a subset of the real axis. In particular, it is shown that both A and C must be translates of hermitian matrices.     

On the range of a generalized derivation

A generalized derivation , is defined by the formula , where and is the Banach algebra of bounded linear operators in a Hilbert space . Sufficient conditions under which and are given. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 111–119.     

A numerical solution of the constrained weighted energy problem

A numerical algorithm is presented to solve the constrained weighted energy problem from potential theory. As one of the possible applications of this algorithm, we study the convergence properties of the rational Lanczos iteration method for the symmetric eigenvalue problem. The constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of the eigenvalues, on the distribution of the poles, and on the ratio between the size of the matrix and the number of iterations. Our algorithm gives the possibility of finding the boundary of this region in an effective way.We give numerical examples for different distributions of poles and eigenvalues and compare the results of our algorithm with the convergence behavior of the explicitly performed rational Lanczos algorithm.     

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.     

On the local maxima of a constrained quadratic form

This note presents a brief and partial review of the work of Broom, Cannings and Vickers 1 Broom, M, Cannings, C and Vickers, GT. 1993. On the number of local maximum of a constrained quadratic form. Proceedings of the Royal Society of London, A443. 1993. pp.573–584.  [Google Scholar]. It also presents some simple examples of an extension of the their formalism to non-symmetric matrices.     

Isometries of a generalized numerical radius

For 0<q<1, the q-numerical range is defined on the algebra M_n of all n×n complex matrices by

W_q(A)={x^∗Ay:x,y∈Cⁿ,∥x∥=∥y∥=1,〈y,x〉=q}.     

Note on a paper of thompson: the congruence numerical range

Given n×n Complex matrices A, Cdefine the C-congruence numerical range of A to be the set [ILM0001]. R.C. Thompson has characterized R_C(A) when [ILM0002] are fixed complex numbers. In this note. we obtain some analogous results about R_t(A) when C is skew symmmetric and a simple proof of the result of Thompson is given.Moreover, we characterize a certain set of partial off diagonals under congruence unitary transformation.     

A note on a generalized form of the Laplacian and of sub-Laplacians

In this note we prove a recent conjecture of Hasson [11]: we show that, for a locally integrable function u, a sufficient condition to be harmonic is that $ \lim\limits_{r\to 0^+} r^{-2}(M_{r}u-u) = 0 $ in the weak sense of distributions (M _r being the averaging operator on balls of radius r). We also extend this and other results to the setting of sub-Laplacians on Carnot groups.Investigation supported by University of Bologna. Funds for selected research topics.     

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 constrained interval approach to the generalized distance geometry problem

This paper presents a new approach to the generalized distance geometry problem, based on a model that uses constraint interval arithmetic. In addition to theoretical results, we give some computational experiments that illustrate the better performance of the proposed approach, compared to others from the literature.     

A language-driven generalized numerical database translator

Numerical databases arise in many scientific applications to keep track of large dense and sparse matrices, residing on secondary devices in matrix compact data representation. This paper describes a language-driven generalized data translator for translating any numerical database from one matrix compact data representation to another. Our approach is to describe any matrix compact data representation by a physical schema and any numerical database and its mapping to storage by data language facilities. The languages are processed by a Generalized Syntax-Directed Translation Scheme (GSDTS) to automatically generate FORTRAN conversion programs which become the major modules of the translator.