It is proved that a commutative algebra $A$ of operators in a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when } \varepsilon\searrow 0,$$ where $\|\cdot\|_e$ is the essential norm. This implies the existence of an invariant subspace for every commutative family of essentially selfadjoint operators in a real Hilbert space.

Работа содержит обзор основных результатов теории инвариантных подпространств, которые имеют прямую или опосредованную связь с известным списком проблем теории операторов, составленным П. Халмошем в 1970 г. Для большинства результатов приводятся краткие доказательства, схемы или основные идеи доказательств. Библиография: 124 названия.

Изучены закономерности превращения этилена в условиях реакции окислительной конденсации метана. Показано, что в пустом реакторе при температурах выше 740°С с высокой скоростью протекает окисление этилена, основным продуктом которого является монооксид углерода. Заполнение реактора как инертным материалом – кварцем, так и катализатором NaWMn/SiO2 приводит к значительному снижению скорости превращения этилена. Установлено, что добавление метана в реакционную смесь резко снижает скорость превращения этилена и приводит к увеличению содержания С3-углеводородов в продуктах реакции. Исследованы кинетические закономерности окисления этилена с добавками метана на катализаторе NaWMn/SiO2.

Исследована кинетика окислительной конденсации метана (ОКМ) в присутствии катализаторов La/MgO и NaWMn/SiO2 в реакторе проточного типа при малых конверсиях реагентов. Показано, что, несмотря на различие в составе и свойствах исследуемых катализаторов, в обоих случаях кинетические закономерности образования этана из метана и этилена из этана могут быть описаны в рамках окислительно-восстановительной модели Марса–ван-Кревелена. Закономерности побочных реакций, ведущих к образованию оксидов углерода, отличаются от закономерностей целевых реакций превращения метана в этан и этана в этилен. Для каждого катализатора определены кинетические параметры, необходимые для численного моделирования процесса ОКМ.

The effect of water onto the rate and selectivity of methane and ethane oxidation in the conditions of oxidative coupling of methane (OCM) is studied. The effect of water strongly depends on the OCM catalyst composition: whereas PbOX/Al2O3 undergoes an irreversible deactivation, no effect of water is observed on both reaction rate and OCM selectivity over La/MgO. Over NaWMn/SiO2 catalyst both rate of reaction and selectivity are enhanced by water addition to the feed at low conversions during methane oxidation. In the case of ethane oxidation, the rate of reaction is strongly affected by water addition, whereas selectivity to ethylene does not change at equal conversions. At increasing concentration of water in the feed gas its relative effect onto the methane oxidation substantially decreases, so the increasing concentration of water above ~8 vol% does not further enhance the rate of methane oxidation. The observed effects are explained by the participation of water in the active site turnover, namely in its re-oxidation by shifting the prevailing re-oxidation route from filling surface oxygen vacancies to oxidative dehydrogenation of surface hydroxy-groups.

A very simple, short and self-contained proof is presented of Burnside's Theorem that every proper algebra of matrices over an algebraically closed field has a non-trivial invariant subspace.

We prove that if the Bishop–Phelps Theorem is correct for a uniform dual algebra R of operators in a Hilbert space, then the algebra R is selfadjoint.

We show that any weakly closed algebra of bounded operators acting on a Banach space and different from the algebra of all bounded operators admits positive vector-functionals continuous in the essential operator norm.

In the present work we prove theorem on existence of invariant subspaces for a class of operators in a space with indefinite metric.

Let T be an arbitrary linear bounded operator on a complex Banach space B. As usual, we denote by ρ(T) the resolvent set of T, i.e., ρ(T) is the set of λ ∈ C such that the resolvent R λ(T) = (T - λI)y-1 exists in the algebra of all linear bounded operators on B. This set is open and R λ(T) is an analytic operator function on it. The spectrum σ(T) = C/ρ(T) is compact.

Let X be a real or complex normed space, A be a linear operator in the space X, and x is an element of X. We put E(X, A, x) = min{l : l > 0, parallel to A(l)x parallel to not equal parallel to x parallel to}, or 0 if parallel to A(k)x parallel to = parallel to x parallel to for all integer k > 0. Then let E(X, A) = sup(x), E(X, A, x) and E(X) = sup(A) E(X, A). If dim X greater than or equal to 2 then E(X) greater than or equal to dim X + 1. A space X is called E-finite if E(X) < infinity. In this case dim X < infinity, and we set dim X = n.The main results are following, If X is polynomially normed of a degree p, then it is E-finite; moreover, E(X) less than or equal to C-n+p-1(p) (over R), and E(X) less than or equal to (C-n+p/2-1(p/2))(2) (over C). If X is Euclidean complex, then n(2) - n + 2 less than or equal to E(X) less than or equal to n(2) - 1 for n greater than or equal to 3; in particular, E(X) = 8 if n = 3. Also, E(X) = 4 if n = 2. If X is Euclidean real, then [n/2](2) - [n/2] + 2 less than or equal to E(X) less than or equal to n(n + 1)/2, and E(X) = 3 if n = 2. Much more detailed information on E-numbers of individual operators in the complex Euclidean space is obtained. If A is not nilpotent, then E(X, A) less than or equal to 2ns - s(2), where s is the number of nonzero eigenvalues. For any operator A we prove that E(X, A) less than or equal to n(2) - n + t, where t is the number of distinct moduli of nonzero nonunitary eigenvalues. In some cases E-numbers are ''small'' and can be found exactly. For instance, E(X, A) less than or equal to 2 if A is normal, and this bound is achieved. The topic is closely connected with some problems related to the number-theoretic trigonometric sums.

Journal Article Spectral Properties of Some Integral Operators Arising in Potential Theory Get access J. M. Anderson, J. M. Anderson Department of Mathematics University College, London, WC1E 6B7 UNITED KINGDOM Search for other works by this author on: Oxford Academic Google Scholar D. Khavinson, D. Khavinson Department of Mathematics University of Arkansas, Fayetteville, Arkansas 72701 U.S.A. Search for other works by this author on: Oxford Academic Google Scholar V. Lomonosov V. Lomonosov Department of Mathematics Kent State University, Kent, Ohio 44242 U.S.A. Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 43, Issue 4, December 1992, Pages 387–407, https://doi.org/10.1093/qmathj/43.4.387 Published: 01 December 1992 Article history Received: 23 October 1990 Published: 01 December 1992

Click to increase image sizeClick to decrease image size1980 Mathematics Subject Classification (1985 Revision): 46E40

The classical Burnside’s Theorem guarantees in a finite dimensional space the existence of invariant subspaces for a proper subalgebra of the matrix algebra. In this paper we give an extension of Burnside’s Theorem for a general Banach space, which also gives new results on invariant subspaces.