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.

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.

A holomorphic map ψ of the unit disk ito itself induces an operator Cψ on holomorphic functions by composition. We characterize bounded and compact composition operators Cψ on Qp spaces, which coincide with the BMOA for p = 1 and Bloch spaces for p > 1. We also give boundedness and compactness characterizations of Cψ from analytic function space X to Qp spaces, X = Dirichlet space D, Bloch space B or B0 = {f: f′ ∈ H∞}.

We deal with the dual Banach algebras for a locally compact group G. We investigate compact left multipliers on , and prove that the existence of a compact left multiplier on is equivalent to compactness of G. We also describe some classes of left completely continuous elements in .

Suppose that σ:𝔐→𝔐 is an ultraweakly continuous surjective *-linear mapping and d:𝔐→𝔐 is an ultraweakly continuous *-σ-derivation such that d(I) is a central element of 𝔐. We provide a Kadison–Sakai-type theorem by proving that ℌ can be decomposed into and d can be factored as the form , where δ:𝔐→𝔐 is an inner *-σ𝔎-derivation, Z is a central element, 2τ=2σ𝔏 is a *-homomorphism, and σ𝔎 and σ𝔏 stand for compressions of σ to 𝔎and 𝔏 , respectively.

In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identify the isometries and spectra of a particular class of weighted composition operators on the Bloch space.

We consider in this paper the question of when the finite sum of products of two Toeplitz operators is a finite-rank perturbation of a single Toeplitz operator on the Hardy space over the unit disk. A necessary condition is found. As a consequence we obtain a necessary and sufficient condition for the product of three Toeplitz operators to be a finite-rank perturbation of a single Toeplitz operator.

Given a positive continuous function $\mu $ on the interval $0\,<\,t\,\le \,1$ , we consider the space of so-called $\mu $ -Bloch functions on the unit ball. If $\mu \left( t \right)\,=\,t$ , these are the classical Bloch functions. For $\mu $ , we define a metric $F_{z}^{\mu }\left( u \right)$ in terms of which we give a characterization of $\mu $ -Bloch functions. Then, necessary and sufficient conditions are obtained in order that a composition operator be a bounded or compact operator between these generalized Bloch spaces. Our results extend those of Zhang and Xiao.

We consider the problem of determining for which square integrable functions $f$ and $g$ on the polydisk the densely defined Hankel product ${{H}_{f}}\,H_{g}^{*}$ is bounded on the Bergman space of the polydisk. Furthermore, we obtain similar results for the mixed Haplitz products ${{H}_{g}}\,{{T}_{{\bar{f}}}}$ and ${{T}_{f}}\,H_{g}^{*}$ , where $f$ and $g$ are square integrable on the polydisk and $f$ is analytic.

In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this result is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply our theorem to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.

We introduce the spaces Vℬp(X) (respectively 𝒱ℬp(X)) of the vector measures ℱ:Σ→X of bounded (p,ℬ)-variation (respectively of bounded (p,ℬ)-semivariation) with respect to a bounded bilinear map ℬ:X×Y →Z and show that the spaces Lℬp(X) consisting of functions which are p-integrable with respect to ℬ, defined in by Blasco and Calabuig [‘Vector-valued functions integrable with respect to bilinear maps’, Taiwanese Math. J. to appear], are isometrically embedded in Vℬp(X). We characterize 𝒱ℬp(X) in terms of bilinear maps from Lp′×Y into Z and Vℬp(X) as a subspace of operators from Lp′(Z*) into Y*. Also we define the notion of cone absolutely summing bilinear maps in order to describe the (p,ℬ)-variation of a measure in terms of the cone-absolutely summing norm of the corresponding bilinear map from Lp′×Y into Z.

Let ϕ:D→D and ψ:D→ℂ be analytic maps. These induce a weighted composition operator ψCϕ acting between weighted Bloch type spaces. Under some assumptions on the weights we give a necessary as well as a sufficient condition when such an operator is continuous.

Let , let G and H belocally compact groups and let ω be a continuous homomorphism of G into H. We prove, if G is amenable, the existence of a linear contraction of the Banach algebra CVp(G) of the p-convolution operators on G into CVp(H) which extends the usual definition of the image of a bounded measure by ω. We also discuss the uniqueness of this linear contraction onto important subalgebras of CVp(G). Even if G and H are abelian, we obtain new results. Let Gd denote the group G provided with a discrete topology. As a corollary, we obtain, for every discrete measure, , for Gd amenable. For arbitrary G, we also obtain . These inequalities were already known for p=2 . The proofs presented in this paper avoid the use of the Hilbertian techniques which are not applicable to . Finally, for Gd amenable, we construct a natural map of CVp (G)into CVp (Gd) .

T. A. Gillespie showed that on a Hilbert space the sum of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. A longstanding question asked whether this might still hold true for operators on Lp spaces for . We show here that this conjecture is false. Indeed for a large class of reflexive spaces, the above property characterizes Hilbert space.

We consider differences of composition operators between given weighted Banach spaces or Hv0 of analytic functions with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. Some examples illustrate our results.

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.