A discreteness criterion for the spectrum of a quasielliptic operator

For the spectrum of the operator $$u = \sum\nolimits_{j = 1}^n {( - 1)^{m_j } D_j^{2m_j } u + q(x)u,} $$ to be discrete, where the m_j are arbitrary positive integers such that \(\sum\nolimits_{j = 1}^n {\tfrac{1}{{2m_j }}< 1} \) , and q(x) ≥ 1, it is necessary and sufficient that \(\int\limits_K {q (x) dx \to \infty } \) , when the cube K tends to infinity while preserving its dimensions.     

A criterion for an abelian variety to be simple

In this note we give a numerical criterion that expresses the condition that an abelian variety be simple in terms of an invariant that is closely related to the s-invariant of Ein-Cutkosky-Lazarsfeld. Received: 1 July 2007     

Univalence criterion and convexity for an integral operator

For analytic functions f and g in the open unit disk U, a new integral operator I₁(f,g)(z) is introduced. The main object of this paper is to obtain a univalence condition and the order of convexity for the integral operator I₁(f,g)(z).     

A criterion for a process to be prime

A criterion for a measure preserving transformation to be prime is given. The criterion allows the transformation to have uncountable centralizer.Research supported by NSF-MCS-80-26540.     

A criterion for Banach manifolds to be finite-dimensional

We extend the results obtained in [1] to the case of arbitrary Banach spaces and manifolds. We give an example of a continuous bijective mapping with discontinuous inverse which acts in a Banach space and differs from the identical mapping only in an open unit ball. A criterion for a Banach manifold to be finite-dimensional is established in terms of the continuity of inverse operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1712–1713, December, 1995.     

A semi-iterative method for real spectrum singular linear systems with an arbitrary index

In this paper we develop a semi-iterative method for computing the Drazin-inverse solution of a singular linear system Ax = b, where the spectrum of A is real, but its index (i.e., the size of its largest Jordan block corresponding to the eigenvalue zero) is arbitrary. The method employs a set of polynomials that satisfy certain normalization conditions and minimize some well-defined least-squares norm. We develop an efficient recursive algorithm for implementing this method that has a fixed length independent of the index of A. Following that, we give a complete theory of convergence, in which we provide rates of convergence as well. We conclude with a numerical application to determine eigenprojections onto generalized eigenspaces. Our treatment extends the work of Hanke and Hochbruck (1993) that considers the case in which the index of A is 1.     

Instability spectrum of an operator bundle

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

A criterion for Banach manifolds with a separable model to be finite-dimensional

We give an example of a continuous bijective mapping with a discontinuous inverse which acts in a separable Banach space and differs from the identical mapping only in an open unit ball. A criterion for Banach manifolds with a separable model to be finite-dimensional is established in terms of the continuity of inverse operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1099–1103, August, 1994.     

Conditions for the negative spectrum of the Schrodinger equation operator to be discrete and finite

Let H be a separable Hilbert space and let H₁ be a Hilbert space whose elements are vector functions f(x) (0x<) with domain H. Define the scalar product in H₁ to be (f(x), g(x))₁= . This paper concerns the study of the negative spectrum of the operatorl(y) =–y +Q(x)y, y'(0)–hy(0)=0, y(x) H₁, where Q(x) is a strongly measurable operator function in H which is completely continuous for almost all x, and where h is a fixed completely continuous operator in H. There is 1 reference.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 531–538, November, 1967.     

Noncontinuity of spectrum for the adjoint of an operator

This paper deals with the connection between continuity of spectrum at an element of the Banach algebra of all bounded linear operators on a Banach space and at the adjoint of . In particular, we show that, if is not reflexive, the spectrum function may be continuous at and discontinuous at .