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]).

Let M be an invariant subspace of L2 (T2) on the bidisc. V1 and V2 denote the multiplication operators on M by coordinate functions z and ω, respectively. In this paper we study the relation between M and the commutator of V1 and , For example, M is studied when the commutator is self-adjoint or of finite rank.

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, ∞).

We construct a functional calculus, g → g(A), for functions, g, that are the sum of a Stieltjes function and a nonnegative operator monotone function, and unbounded linear operators, A, whose resolvent set contains (−∞, 0), with {‖r(r + A)−1‖ ¦ r > 0} bounded. For such functions g, we show that –g(A) generates a bounded holomorphic strongly continuous semigroup of angle θ, whenever –A does.

We show that, for any Bernstein function f, − f(A) generates a bounded holomorphic strongly continuous semigroup of angle π/2, whenever − A does.

We also prove some new results about the Bochner-Phillips functional calculus. We discuss the relationship between fractional powers and our construction.

Well-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

Let T1, i = 1, 2 be measurable transformations which define bounded composition operators C Ti on L2 of a σ-finite measure space. Let us denote the Radon-Nikodym derivative of with respect to m by hi, i = 1, 2. The main result of this paper is that if and are both M-hyponormal with h1 ≤ M2(h2 o T2) a.e. and h2 ≤ M2(h1 o T1) a.e., then for all positive integers m, n and p, []* is -hyponormal. As a consequence, we see that if is an M-hyponormal composition operator, then is -hyponormal for all positive integers n.

Let X be a completely regular Hausdorff space, let V be a system of weights on X and let T be a locally convex Hausdorff topological vector space. Then CVb(X, T) is a locally convex space of vector-valued continuous functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper we characterize multiplication operators on the space CVb(X, T) induced by operator-valued mappings and then obtain a (linear) dynamical system on this weighted function space.

Let S(n) be a unilateral shift operator on a Hilbert space of multiplicity n. In this paper, we prove a generalization of the theorem that if S(1) is unitarily equivalent to an operator matrix form relative to a decomposition ℳ ⊕ N, then E is in a certain class C0 which will be defined below.

Ahues (1987) and Bouldin (1990) have given sufficient conditions for the strong stability of a sequence (Tn) of operators at an isolated eigenvalue of an operator T. This paper provides a unified treatment of their results and also generalizes so as to facilitate their application to a broad class of operators.

If V is a system of weights on a completely regular Hausdorff space X and E is alocally convex space, then CV0(X, E) and CVb (X, E) are locally convex spaces of vector-valued continuous functions with topologies generated by seminorms which are weighted analogues of the supremum norm. In this paper we characterise multiplication operators on these spaces induced by scalar-valued and vector-valued mappings. Many examples are presented to illustrate the theory.

We present a symmetric version of a normed algebra of quotients for each ultraprime normed algebra. In addition, a C*-a1gebra of quotients of an arbitrary C*-a1gebra is introduced.

Suppose λ is an isolated eigenvalue of the (bounded linear) operator T on the Banach space X and the algebraic multiplicity of λ is finite. Let Tn be a sequence of operators on X that converge to T pointwise, that is, Tnx → Tx for every x ∈ X. If ‖(T − Tn)Tn‖ and ‖Tn(T − Tn)‖ converge to 0 then Tn is strongly stable at λ.

Spectrality and prespectrality of elementary operators , acting on the algebra B(k) of all bounded linear operators on a separable infinite-dimensional complex Hubert space K, or on von Neumann-Schatten classes in B(k), are treated. In the case when (a1, a2, …, an) and (b1, b2, …, bn) are two n—tuples of commuting normal operators on H, the complete characterization of spectrality is given.

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).

The lack of completeness with respect to the semivariation norm, of the space of Banach space valued functions, Pettis integrable with respect to a measure μ, often impedes the direct extension of results involving integral representations, true in the finite-dimensional setting, to the general vector space setting. It is shown here that the space of functions with values in a space Y, μ-Archimedes integrable in a Banach space X embedded in Y, is complete with respect to convergence in semivariation, provided the embedding from X into Y is completely summing. The result is applied to the case when Y is a conuclear space, in particular, when X is a function space continuously included in a space of distributions.

We prove a version of the Feller-Miyadera-Phillips theorem characterizing the infinitesimal generators of positive C0-semigroups on ordered Banach spaces with normal cones. This is done in terms of N(A) as well as the canonical half-norms of Arendt Chernoff and Kato defined by N(a) = inf{‖b‖ |b ≥ a}, where N(A) = sup{N(Aa) |N(a) ≶ 1} for operator A. A corresponding result on –semigroups is also given.

We give algebraic criteria for distinguishing composition operators among all continuous linear operators on spaces of continuous functions with topologies generated by seminorms that are weighted analogues of the supremum norm. In another direction, we also characterize those self maps of the underlying topological space which induce composition operators on such weighted spaces, as well as determine conditions on these self maps which correspond to various basic properties of the induced composition operator. Our results are applied to a question concerning translation invariance which arises in the context of topological dynamics.

The notion of a scalar operator on a Banach space, in the sense of N. Dunford, is widened so as to cover those operators which can be approximated in the operator norm by linear combinations of disjoint values of an additive and multiplicative operator valued set function, P, on an algebra of sets in a space Ω such that P(Ω) = I, subject to some conditions guaranteeing that this definition is unambiguous. An operator T turns out to be scalar in this sense, if and only if, there exists a (not necessarily bounded) Boolean algebra of bounded projections such that the Banach algebra of operators it generates is semisimple and contains T.

We construct a suitable representation of a C*-algebra that carries single elements to rank one operators. We also prove an abstract spectral theorem for compact elements in the algebra. This leads naturally to an abstract definition of Cp-classes of compact elements in the algebra.

A proof is given of the Euler-Maclaurin sum formula, on a Banach space of differentiable vector-valued functions of bounded exponential growth, using the Laplace transformation. Some related summation formulae are proved by the same methods. Properties of the standard summation operator are proved, namely spectral properties and boundedness, continuity and differentiability results.