Transformations of linear connections

Combining several results on related (or conjugate) connections, defined on banachable fibre bundles, we set up a machinery, which permits to study various transformations of linear connections. Global and local methods are applied throughout. As an application, we get an extension of the classical affine transformations to the context of infinite-dimensional vector bundles. Another application shows that, realising the ordinary linear differential equations (in Banach spaces) as connections, we get the usual transformations of (equivalent) equations. Thus, some classical results on differential equations, such as the Theorem of Floquet, can have a geometric interpretation.     

Testing linear operators

The goal of testing is to determine whether an implementation linear operatorA conforms to a specification linear operatorS within a given error bound for all elements from an input setF. Suppose that an upper boundK on the norm of the difference ofS andA is given a priori. Then it is shown that in general any finite number of tests is inconclusive both in the worst case and on the average. However, the testing problem is still decidable in the limit for an arbitraryK; there is an algorithm of an infinite sequence of test-and-guess such that all but finitely many guesses are correct. On the other hand, if the error bound is relaxed for weak conformance then finite tests suffice even in the worst case and tight lower and upper bounds on the number of tests are derived. The test set is universal; it only depends on the set of valid inputsF. Furthermore, the test elements are on the boundary ofF. Two examples are used to illustrate the approaches and the paper is concluded with comments on two related problems: computation and verification.This work was done while consulting at AT&T Bell Laboratories, and is partially supported by the National Science Foundation grant IRI-92-12597 and the Air Force Office of Scientific Research 91-0347.     

Geometries involving affine connections and general linear connections

Summary Part I of this paper is concerned with the theory of differential invariants of a symmetric affine connection and a general linear connection of theK?nig type. Part II deals with a geometry in which the components of the affine connection areChristoffel symbols, and the general linear connection is of a special sort. This section can be considered in part as an extension of theEinstein-Mayer (5) geometry,     

Gauge-natural forms of Chern-Weil type

We determine all gauge-natural forms on a principal fiber bundle with values in an arbitrary associated vector bundle, which generalize the classical Chern-Weil forms.     

Invariant linear connections on homogeneous symplectic varieties

We find all homogeneous symplectic varieties of connected semisimple Lie groups that admit an invariant linear connection.The research was supported by the grant INTAS-OPEN-97-1570 of the INTAS Foundation.     

Some spectral properties of a bundle of linear operators in Banach space

The spectral properties of a bunchA–B, D(A)D(B), of linear closed densely defined operators in Banach space are considered. Our main result is a theorem to the effect that the spectrum of the bunch can be expanded with respect to a pair of direct sums; the theorem generalizes the celebrated theorem of Riesz concerning the expansion of the spectrum of an operator.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 847–857, December, 1977.     

Geometric connections and geometric Dirac operators on contact manifolds

We construct some natural metric connections on metric contact manifolds compatible with the contact structure and characterized by the Dirac operators they determine. In the case of CR manifolds these are invariants of a fixed pseudo-hermitian structure, and one of them coincides with the Tanaka-Webster connection.     

Some results on b-AM-compact operators

We show that for positive operator B : E → E on Banach lattices, if there exists a positive operator S : E → E such that:1.SB ≤ BS;2.S is quasinilpotent at some x₀ > 0; 3.S dominates a non-zero b-AM-compact operator, then B has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM- compact-friendly operator which is quasinilpotent at some x₀ > 0 has a non-trivial closed invariant ideal.     

Fuzzy bounded linear operators

In this paper, a notion of boundedness of a linear operator from a fuzzy normed linear space to another fuzzy normed linear space is introduced and two types (strong and weak) of fuzzy bounded linear operators are defined. Relation between fuzzy continuity and fuzzy boundedness is studied. Definitions of fuzzy bounded linear functionals are given and the notions of fuzzy dual spaces are developed. The Hahn-Banach theorem, the Open mapping theorem, the Closed graph theorem and the Uniform boundedness principle theorem are established.     

Outer preserving linear operators

A natural question about linear operators on the Hilbert-Hardy space is answered, motivated by work in geophysical imaging. Namely, which bounded linear operators on the Hardy space preserve the set of all shifted outer functions? A complete characterization is determined, which allows an explicit construction of all such operators. Every operator that preserves the set of shifted outer functions is necessarily a product-composition operator, consisting of composition with a shifted outer function followed by multiplication with a (possibly different) shifted outer function. Such operators represent important physical processes, including the propagation of seismic wave energy through the earth. Applications to seismic imaging are briefly discussed.     

Fuzzy compact linear operators

In this paper, we introduce fuzzy compact linear operators between fuzzy normed spaces and investigate some important general properties of them.     

Interpolation of linear operators

The survey is devoted to the modern state of the theory of interpolation of linear operators acting in Banach spaces. Principal attention is devoted to real and complex methods and applications of the theory of interpolation to analysis.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 24, pp. 3–164, 1986.