Algebraic and essentially algebraic composition operators onC(X)

Summary Given a compact Hausdorff spaceX, we may associate with every continuous mapa: X X a composition operatorC _a onC(X) by the rule(C _a f)(x) = f(a(x)). We describe all self-mapsa for whichC _a is an algebraic operator or an essentially algebraic operator (i.e. an operator algebraic modulo compact operators), determine the characteristic polynomialp _a (z) and the essentially characteristic polynomialq _a (z) in these cases and show how the connectivity ofX may be characterized in terms of the quotientsp _a (z)/q _a (z). Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer.     

Weak compactness and compactness of dominated operators onC(Ω,X)

In questo lavoro presentiamo alcuni risultati relativi alla debole compattezza e alla compattezza degli operatori dominati sugli spaziC(Ω,X).     

Peripheral spectrum of positive quasi-compact operators onC 0(X)

LetX be a locally compact space, andT, a quasi-compact positive operator onC ₀(X), with positive spectral radius,r. Then the peripheral spectrum ofT is a finite set of poles containingr, and the residue of the resolvent ofT at each peripheral pole is of finite rank. Using the concept of closed absorbing set, we develop an iterative process that gives the order,p, ofr, some special bases of the algebraic eigenspaces ker(T-r) ^p and ker(T ^*-r) ^p , and finally the dimension of the algebraic eigenspace associated to each peripheral pole.     

Norm-attaining functionals onC(Q, X)

Functionals (vector measures) defined on the spaceC(Q, X) of continuous abstract functions (whereQ is a compact Hausdorff space andX is a Banach space) and attaining their norm on the unit sphere are considered. A characterization of such functionals is given in terms of the Radon-Nikodym derivative of the vector measure with respect to the variation of the measure and in terms of analogs of the derivative. Applications to the characterization of finite-codimensional subspaces with the best approximation property are given. Similar results are obtained for the spaceB(Q, Σ, X) of uniform limits of simple functions. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 45–56, January, 1997. Translated by V. E. Nazaikinskii     

Barrelledness conditions onC 0 (E)

Some conditions of barrelledness are considered and studied on the spaceC ₀(E), defined as follows: IfE is a real or complex Hausdorff locally convex space and \(P_E \) is a saturated family of seminorms, defining the original topology ofE, then the vector space of all the sequences \(\bar f = \left\{ {\bar f(n): n \in \mathbb{N}} \right\}\) inE, convergent to zero, provided with the locally convex topology $$\bar p(\bar f) = sup\left\{ {p (\bar f(n)): n \in \mathbb{N}} \right\}p \in P_E $$ is defined as the spaceC ₀(E). The main result of the paper is the following characterization:C ₀(E) is quasibarrelled (see [3], p. 367) if and only if,E is quasibarrelled and the strong dual ofE has property (B) (see [5], p. 30, for definition). We obtain. as a consequence, commutativity properties of the operatorC ₀, acting on certain inductive limits (3.3 Theorem). We also prove thatC ₀ does not commute with uncountably strict inductive limits. In particular, there are ultrabornological spacesE for whichC ₀(E) is not quasibarrelled. 3.1. Example provides a complete?-tensor product of two complete ultrabornological spaces which is not quasibarrelled.     

Coboundaries of irreducible markov operators onC(K)

LetK be a compact Hausdorff space, and letT be an irreducible Markov operator onC(K). We show that ifgεC(K) satisfies sup _N ‖Σ _j ^=0N T ^j g‖<∞, then (and only then) there existsfεC(K) with (I − T)f=g. Generalizing the result to irreducible Markov operator representations of certain semi-groups, we obtain that bounded cocycles are (continuous) coboundaries. For minimal semi-group actions inC(K), no restriction on the semi-group is needed.     

Coarse version of the Banach-Stone theorem

We show that if there exists a Lipschitz homeomorphism T between the nets in the Banach spaces C(X) and C(Y) of continuous real valued functions on compact spaces X and Y, then the spaces X and Y are homeomorphic provided . By l(T) and l(T−1) we denote the Lipschitz constants of the maps T and T−1. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is . We also estimate the distance of the map T from the isometry of the spaces C(X) and C(Y).     

Regular and singular Hermitian operators

Let A be a closed Hermitian operator, let be the orthogonal complement of the domain of definition of A, and let be the defect subspace. An operator A is called regular if the orthogonal projection of on is closed. Criteria for regularity are established.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 197–203, August, 1970.     

Stable bundles, representation theory and Hermitian operators

We give an interpretation and a solution of the classical problem of the spectrum of the sum of Hermitian matrices in terms of stable bundles on the projective plane.