Instability spectrum of an operator bundle

Translated from Matematicheskie Zametki, Vol. 49, No. 4, pp. 88–94, April, 1991.     

Interpolation of operator ideals with an application to eigenvalue distribution problems

We derive results on the interpolation of complete quasinormed operator ideals, mainly for the absolutelyp-summing and thes-number idealsS _p ^s defined by Pietsch. By estimating theK-Functional of Peetre, we get that the interpolation ideal (S _p1 ^s ,S _p2 ^s ),_,p is contained inS _p ^s and is even equal to it in the case of the approximation numbers. A similar fact is proved for absolutely (p, q)-summing operators, interpolating the first index. We show further that the absolutelyp-summing operators onc ₀ are contained in the complex interpolation space ( _p1 (c _o), _p2 (c _o))_[].The previous results are then applied to prove summability properties for the eigenvalues of operators in Banach spaces, which are products ofS _p1 ^s -type and absolutelyp _j -summing operators. Roughly speaking, the summability order is the harmonic sum of thep _i - andp _j -indices, wherep _j 2. In the case of Hilbert spaces, this reduces to the well-known Weyl-inequality. The method uses an abstract interpolation estimate for ideal quasinorms which may be useful also for other operator ideals.     

Characterization of the spectrum of regular boundary value problems for the Sturm-Liouville operator

On the interval (0, τ), we consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L ₂(0, τ) and with regular (but not strengthened-regular) boundary conditions. Under certain additional assumptions, we establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.     

Functional calculus and its application to operator algebras

In this paper we will prove that any unital isometric representation ofH () with finitely connected domain has property (A ₁(r)) for somer1, which generalizes the same conclusion in [2] and [8].     

On an operator approach to interpolation problems for Stieltjes functions

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapov's method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered.     

Estimate for the remainder in the Weyl asymptotics of the spectrum of the Maxwell operator in Lipschitz domains

We study the asymptotics of the spectrum of the Maxwell operator M in a bounded Lipschitz domain W ì \mathbbR³ \Omega \subset {\mathbb{R}^3} under the condition of the perfect conductivity of the boundary ∂Ω. We obtain the following estimate for the remainder in the Weyl asymptotic expansion of the counting function N(λ,M) of positive eigenvalues of the Maxwell operator M:

A criterion for an operator with real spectrum to be of scalar-type

Using a vector version of the Bochner—Schoenberg test which characterizes the Fourier—Stieltjes transforms of measures, a criterion is presented, in terms of the resolvent map, which determines when a linear operator with real spectrum acting in a locally convex space is a scalar-type spectral operator. This is an extension of a recent result of S. Kantorovitz.     

On the spectrum of an odd-order differential operator

Operators generated by a differential expression on a finite closed interval are considered. It is shown that, for any odd integer n, there exist differential operators of order n whose spectrum fills the whole complex plane.     

A nonconforming finite element method to compute the spectrum of an operator relative to the stability of a plasma in toroidal geometry

Summary In this paper we describe a nonconforming finite element method to compute the MHD spectrum of a plasma in a toroïdal configuration. We show that this method leads to a good approximation of the spectrum.     

Diminishing of the spectrum of an operator under extension

A negative answer is given to the second question posed on page 210 of the book Complex Analysis — 199 Research Problems, Lecture Notes in Math., No. 1043, Springer, Berlin (1984).Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 45, pp. 96–97, 1986.     

A variational principle for block operator matrices and its application to the angular part of the Dirac operator in curved spacetime

The operator associated to the angular part of the Dirac equation in the Kerr-Newman background metric is a block operator matrix with bounded diagonal and unbounded off-diagonal entries. The aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above. In the last section, these analytic bounds are compared with numerical values from the literature.     

On the asymptotic distribution of the spectrum of an operator over irreducible representations of its symmetry group

We show that a representation of a finite group in the eigenfunction space of an elliptic operator defined on a Riemannian manifold and commuting with the effective action of the group is asymptotically a multiple of the regular representation of the group.