We show the existence of invariant measures for Markov-Feller operators defined on completely regular topological spaces which satisfy the classical positivity condition.

We show that the poles of a resolvent coincide with the poles of its weak resolvent up to their orders, for operators on Hilbert space which have some cyclic properties. Using this, we show that a theorem similar to the Mlak theorem holds under milder conditions, if a given operator and its adjoint have cyclic vectors.

The present paper deals with a question of M. Mbekhta concerning partial isometries on Banach spaces.

In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = 1, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille’s results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand’s theorem for m commuting bounded operators, $T_1,...,T_m$, on a Banach space X.

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....