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.

Let ${[X_0,X_1]}_t$, 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_0$ and $X_1$ will act boundedly on each ${[X_0,X_1]}_t$. Let $T_t$ denote such an operator when considered on ${[X_0,X_1]}_t$, and $\sigma \left(T_t\right)$ denote its spectrum. We are motivated by the question of whether or not the map $t\to \sigma \left(T_t\right)$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t\to {\left(\sigma \left(T_t\right)\right)}^{\wedge }$ (polynomially convex hull) and $t\to {\partial }_e\left(\sigma \left(T_t\right)\right)$ (boundary of the polynomially convex...

Let G be a locally compact Polish group with an invariant metric. We provide sufficient and necessary conditions for the existence of a compact set A ⊆ G and a sequence $g_n\in G$ such that ${\mu }^{\ast n}\left(g_{n}A\right)\equiv 1$ for all n. It is noticed that such measures μ form a meager subset of all probabilities on G in the weak measure topology. If for some k the convolution power ${\mu }^{\ast k}$ has nontrivial absolutely continuous component then a similar characterization is obtained for any locally compact, σ-compact, unimodular, Hausdorff topological...

2000 Mathematics Subject Classification: 47A65, 45S78.Connection of characteristic functions S(z) of nonunitary operator T with the functions of Caratheodori class is established. It was demonstrated that the representing measures from integral representation of the function of Caratheodori's class are defined by restrictions of spectral measures of unitary dilation, of a restricted operator T on the corresponding defect subspaces.