Let 𝒜 be a Banach algebra over ℂ with unit 1 and 𝑓: ℂ → ℂ an entire function. Let 𝐟: 𝒜 → 𝒜 be defined by 𝐟(a) = 𝑓(a) (a ∈ 𝒜), where 𝑓(a) is given by the usual analytic calculus. The connections between the periods of 𝑓 and the periods of 𝐟 are settled by a theorem of E. Vesentini. We give a new proof of this theorem and investigate further properties of periods of 𝐟, for example in C*-algebras.

Let $T=T\left(u\right):u\in {ℝ}_d^{+}$ be a strongly continuous d-dimensional semigroup of linear contractions on $L_1((\Omega ,\Sigma ,\mu );X)$, where (Ω,Σ,μ) is a σ-finite measure space and X is a reflexive Banach space. Since $L_1((\Omega ,\Sigma ,\mu );X)*=L_{\infty }((\Omega ,\Sigma ,\mu );X*)$, the adjoint semigroup $T*=T*\left(u\right):u\in {ℝ}_d^{+}$ becomes a weak*-continuous semigroup of linear contractions acting on $L_{\infty }((\Omega ,\Sigma ,\mu );X*)$. In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), $u\in {ℝ}_d^{+}$, has a contraction majorant P(u) defined on $L_1((\Omega ,\Sigma ,\mu );ℝ)$, that is, P(u) is a positive linear contraction on $L_1((\Omega ,\Sigma ,\mu );ℝ)$ such that $\Vert T\left(u\right)f\left(\omega \right)\Vert \le P\left(u\right)\Vert f\left(·\right)\Vert \left(\omega \right)$ almost everywhere...

Antilinear operators on a complex Hilbert space arise in various contexts in mathematical physics. In this paper, an analogue of the Weyl-von Neumann theorem for antilinear self-adjoint operators is proved, i.e. that an antilinear self-adjoint operator is the sum of a diagonalizable operator and of a compact operator with arbitrarily small Schatten p-norm. On the way, we discuss conjugations and their properties. A spectral integral representation for antilinear self-adjoint operators is constructed....

We give several conditions for (A,m)-expansive operators to have the single-valued extension property. We also provide some spectral properties of such operators. Moreover, we prove that the A-covariance of any (A,2)-expansive operator T ∈ ℒ(ℋ ) is positive, showing that there exists a reducing subspace ℳ on which T is (A,2)-isometric. In addition, we verify that Weyl's theorem holds for an operator T ∈ ℒ(ℋ ) provided that T is (T*T,2)-expansive. We next study (A,m)-isometric operators as a special...

Let A be a closed linear operator acting in a separable Hilbert space. Denote by co(A) the closed convex hull of the spectrum of A. An estimate for the norm of f(A) is obtained under the following conditions: f is a holomorphic function in a neighbourhood of co(A), and for some integer p the operator $A^p-{(A*)}^p$ is Hilbert-Schmidt. The estimate improves one by I. Gelfand and G. Shilov.

For a bounded and sectorial linear operator V in a Banach space, with spectrum in the open unit disc, we study the operator $Ṽ={\int }_0^{\infty }d\alpha V^{\alpha }$. We show, for example, that Ṽ is sectorial, and asymptotically of type 0. If V has single-point spectrum 0, then Ṽ is of type 0 with a single-point spectrum, and the operator I-Ṽ satisfies the Ritt resolvent condition. These results generalize an example of Lyubich, who studied the case where V is a classical Volterra operator.

It is proved that a doubly stochastic operator P is weakly asymptotically cyclic if it almost overlaps supports. If moreover P is Frobenius-Perron or Harris then it is strongly asymptotically cyclic.

Commuting multi-contractions of class $C_{0·}$ and having a regular isometric dilation are studied. We prove that a polydisc contraction of class $C_{0·}$ is the restriction of a backwards multi-shift to an invariant subspace, extending a particular case of a result by R. E. Curto and F.-H. Vasilescu. A new condition on a commuting multi-operator, which is equivalent to the existence of a regular isometric dilation and improves a recent result of A. Olofsson, is obtained as a consequence.

We show that every class A operator has a scalar extension. In particular, such operators with rich spectra have nontrivial invariant subspaces. Also we give some spectral properties of the scalar extension of a class A operator. Finally, we show that every class A operator is nonhypertransitive.