Let L1(ω) be the weighted convolution algebra L1ω(ℝ+) on ℝ+ with weight ω. Grabiner recently proved that, for a nonzero, continuous homomorphism Φ:L1(ω1)→L1(ω2), the unique continuous extension to a homomorphism between the corresponding weighted measure algebras on ℝ+ is also continuous with respect to the weak-star topologies on these algebras. In this paper we investigate whether similar results hold for homomorphisms from L1(ω) into other commutative Banach algebras. In particular, we prove that for the disc algebra every nonzero homomorphism extends uniquely to a continuous homomorphism which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras A+v on (including the algebra of absolutely convergent Taylor series on ) we prove that every nonzero homomorphism Φ:L1(ω)→A+v extends uniquely to a continuous homomorphism which is also continuous with respect to the weak-star topologies.

A new notion of L(n)-hyponormality is introduced in order to provide a bridge between subnormality and paranormality, two concepts which have received considerable attention from operator theorists since the 1950s. Criteria for L(n)-hyponormality are given. Relationships to other notions of hyponormality are discussed in the context of weighted shift and composition operators.

We investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class 𝒮p are obtained. We also construct a compact composition operator which does not belong to any Schatten class.

We give derivative-free characterizations for bounded and compact generalized composition operators between (little) Zygmund type spaces. To obtain these results, we extend Pavlović’s corresponding result for bounded composition operators between analytic Lipschitz spaces.

We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic (‘double root’) situation. For the model with ‘non-smooth’ matrix entries we obtain the asymptotics of generalized eigenvectors and analyse the spectrum. In addition, we reformulate a very helpful theorem from a paper by Janas and Moszynski in its full generality in order to serve the needs of our method.

We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie in the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-sided estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method and illustrate our results with various numerical experiments.

On a separable C*-algebra A every (completely) bounded map which preserves closed two-sided ideals can be approximated uniformly by elementary operators if and only if A is a finite direct sum of C*-algebras of continuous sections vanishing at ∞ of locally trivial C*-bundles of finite type.

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. We consider discrete Tracy–Widom operators and give sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.

Let n∈ℕ and let A and B be rings. An additive map h:A→B is called an n-Jordan homomorphism if h(an)=(h(a))n for all a∈A. Every Jordan homomorphism is an n-Jordan homomorphism, for all n≥2, but the converse is false in general. In this paper we investigate the n-Jordan homomorphisms on Banach algebras. Some results related to continuity are given as well.

Let 𝔻 be the open unit disc, let v:𝔻→(0,∞) be a typical weight, and let Hv∞ be the corresponding weighted Banach space consisting of analytic functions f on 𝔻 such that . We call Hv∞ a typical-growth space. For ϕ a holomorphic self-map of 𝔻, let Cφ denote the composition operator induced by ϕ. We say that Cφ is a bellwether for boundedness of composition operators on typical-growth spaces if for each typical weight v, Cφ acts boundedly on Hv∞ only if all composition operators act boundedly on Hv∞. We show that a sufficient condition for Cφ to be a bellwether for boundedness is that ϕ have an angular derivative of modulus less than 1 at a point on ∂𝔻. We raise the question of whether this angular-derivative condition is also necessary for Cφ to be a bellwether for boundedness.

We express the operator norm of a weighted composition operator which acts from the Bloch space ℬ to H∞ as the supremum of a quantity involving the weight function, the inducing self-map, and the hyperbolic distance. We also express the essential norm of a weighted composition operator from ℬ to H∞ as the asymptotic upper bound of the same quantity. Moreover we study the estimate of the essential norm of a weighted composition operator from H∞ to itself.

Let φ and ψ be holomorphic self-maps of the unit polydisc Un in the n-dimensional complex space, and denote by Cφ and Cψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators Cφ−Cψ from Bloch space to bounded holomorphic function space in the unit polydisc. The compactness of the difference is also characterized.

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.

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.

One of the useful features of spectral measures which happen to be equicontinuous is that their associated integration maps are bicontinuous isomorphisms of the corresponding L1-space onto their ranges. It is shown here that equicontinuity is not necessary for this to be the case; a somewhat weaker property suffices. This is of some interest in practice since there are many natural examples of spectral measures which fail to be equiconontinuous.

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.

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.

We prove the boundedness of Bergman-type operators on mixed norm spaces Lp·q (φ) for 0 < q < 1 and 0 < p ≤ ∞ of functions on the unit ball of ” with an application to Gleason's problem.

In this note we examine the relationships between a subnormal shift, the measure its moment sequence generates, and those of a large family of weighted shifts associated with the original shift. We examine the effects on subnormality of adding a new weight or changing a weight. We also obtain formulas for evaluating point mass at the origin for the measure associated with the shift. In addition, we examine the relationship between the measure associated with a subnormal shift and those of a family of shifts substantially different from the original shift.

We consider the space L1 (ν, X) of all real functions that are integrable with respect to a measure v with values in a real Fréchet space X. We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of l∞ in the space of all linear continuous operators from a complete DF-space Y to a Fréchet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure ν and the space X that imply that L1 (ν, X) has the Dunford-Pettis property. Applications of these results to Fréchet AL-spaces and Köthe sequence spaces are also given.