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.     

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.     

Certain positive linear operators

Properties (including the approximating ones) are investigated of positive linear operators L_n(f; x) for which the relation $$L_n \left( {\left( {t - x} \right)f\left( t \right); x} \right) = \frac{{\varphi \left( x \right)}}{n}L'_n \left( {f\left( t \right); x} \right)$$ is fulfilled, as well as the properties of operators L_n ^(m)(f;x). The results are applicable, in particular, to Bernstein polynomials, to the operators of Mirak'yan, Baskakov, and others.     

Quaternionic triangular linear operators

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S-spectrum. This allow us to obtain conditions when a non-selfadjoint operator admits a triangular representation.     

Graphs of linear operators

In 2006 the author proposed an algorithm for constructing graphs of difference operators. In this paper, the following question is studied: to which linear operators $ \mathcal{A} In 2006 the author proposed an algorithm for constructing graphs of difference operators. In this paper, the following question is studied: to which linear operators does this algorithm apply? Graphs of difference operators are used to determine the complexity of a sequence in the sense of Arnold, so the algorithm makes it possible to determine the complexity of any sequence. Original Russian Text ? A.I. Garber, 2008, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 64–71.     

Periodic points of positive linear operators and Perron-Frobenius operators

LetC(S) denote the Banach space of continuous, real-valued mapsf:S and letA denote a positive linear map ofC(S) into itself. We give necessary conditions that the operatorA have a strictly positive periodic point of minimal periodm. Under mild compactness conditions on the operatorA, we prove that these necessary conditions are also sufficient to guarantee existence of a strictly positive periodic point of minimal periodm. We study a class of Perron-Frobenius operators defined by

Integral representation of linear operators

In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i, _i) (i=1,2) be spaces of finite measure, and let (T,) be the product of these spaces. Let E be an ideal in the space S(T₁, ₁) of measurable functions (i.e., from |e₁||e₂|, e₁ S (T₁, ₁), e₂E it follows that e₁E). THEOREM 2. Let U be a linear operator from E into S(T₂, ₂). The following statements are equivalent: 1) there exists a-measurable kernel K(t,S) such that (Ue)(S)=K(t,S) e(t)d(t) (eE); 2) if 0e_nE (n=1,2,...) and e_n0 in measure, then (Ue_n)(S) 0 ₂ a.e. THEOREM 3. Assume that the function (t,S) is such that for any eE and for s a.e., the ₂-measurable function Y(S)=(t,S)e(t)d ₁(t) is defined. Then there exists a-measurable function K(t,S) such that for any eE we have (t,S)e(t)d ₁(t)=K(t,S)e(t)d ₁(t) ₁a.e.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 5–14, 1974.     

Bounded linear non-absolutely summing operators

We show that, in certain situations, we have lineability in the set of bounded linear and non-absolutely summing operators. Examples on lineability of the set Πp(E,F)?Ip(E,F) are also presented and some open questions are proposed.     

Comparison of linear differential operators

In this paper, we study the question of the existence of the inequality $$\left\| {Q(D)f} \right\|_{L_q } \leqslant \gamma _0 \left\| {P(D)f} \right\|_{L_p } $$ , where P and Q are algebraic polynomials, D = d/dx, and γ ₀ is independent of the function f. We obtain criteria (necessary and simultaneously sufficient conditions) for the existence of such inequalities for functions on the circle, on the whole line, and on the semiaxis. Besides, for the semiaxis, we obtain an inequality for q = ∞ and any p ≥ 1 with the smallest constant γ ₀.     

Generation of linear evolution operators

This paper is devoted to the problem of generation of evolution operators associated with linear evolution equations in a general Banach space. The stability condition is proposed from the viewpoint of finite difference approximations. It is shown that linear evolution operators can be generated even if the stability condition given here is assumed instead of Kato's stability condition.

     

On positive linear interpolation operators

В этой работе мы даем о бобщение понятия нор мальной системы точек, введен ного Фейером [3]. Наше определ ение включает и случа й бесконечного интерв ала (0, ∞). Доказано, в частности, что систе ма точек 0<x ₁ ⁽ⁿ⁾ /(n)<... _n ⁽ⁿ⁾ <∞ является нормальной в смысле нашего определения тогда и т олько тогда, когда вып олняются оценки — фиксированное чис ло, 0≦?<1. Мы доказываем, что есл и точкиx _k ⁽ⁿ⁾ /(n) являются ну лями многочлена ЛагерраL _n ^(α) (x), то они образуют норма льную систему в том и т олько том случае, когда ?1<α≦0. Мы получаем, таким обр азом, положительный интерполяционный пр оцесс для каждой нормальной системы т очек и устанавливаем теорему сходимости для того с лучая, когда эти точки являются ну лямиL _n ^(α) (x) при — 1相似文献