Let A be a C*-algebra, and let X be a Banach A-bimodule. Johnson [B. E. Johnson, ‘Local derivations on C*-algebras are derivations’, Trans. Amer. Math. Soc. 353 (2000), 313–325] showed that local derivations from A into X are derivations. We extend this concept of locality to the higher cohomology of a C*-algebra and show that, for every , bounded local n-cocycles from A(n) into X are n-cocycles.

Let ϕ and ψ be analytic self-maps of the open unit disk. Each of them induces a composition operator, Cϕ and Cψ respectively, acting between weighted Bergman spaces of infinite order. We show that the difference Cϕ−Cψ is compact if and only if both operators are compact or both operators are not compact and the pseudohyperbolic distance of the functions ϕ and ψ tends to zero if ∣ϕ(z)∣→1 or ∣ψ(z)∣→1.

By Pick's invariant form of Schwarz's lemma, an analytic function B (z) which is bounded by one in the unit disk D = {z: |z| < 1} satisfies the inequality

at each point α of D. Recently, several authors [2, 10, 11] have obtained more general estimates for higher order derivatives. Best possible estimates are due to Ruscheweyh [12]. Below in §2 we use a Hilbert space method to derive Ruscheweyh's results. The operator method applies equally well to operator-valued functions, and this generalization is outlined in §3.

We study linear jump parameter systems of differential and difference equations whose coefficients depend on the state of a semi-Markov process. We derive systems of equations for the first two moments of the random solutions of these jump parameter systems, and illustrate how moment equations can be used in examining their asymptotic stability.

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.

Let Ω be a regular domain in the extended complex plane, i.e., it is a bounded domain and its boundary consists of a finite number of disjoint analytic simple closed curves. Let dm(z) be the Lebesgue area measure on Ω and let ds = dm(z)/ω(z) be the Poincare metric on Ω, a Riemannian metric of negative constant curvature. It may be proved that Ω(z) ≈ Euclidean distance from z to the boundary of Ω (see [8]).

The analytic paracommutators in the periodic case have been studied. Their boundedness, compactness, the Schatten-von Neumann properties and the cut-off phenomena have been proved. These results have been applied to some kind of operators on the Bergman spaces that have cut-off at any p∈(0, ∞).

Let P be an irreducible Markov matrix. Assume:

We study conditions for such a matrix to be nonrecurrent. If P is nonrecurrent we study the invariant vectors of P (invariant column vectors and invariant row vectors).