In this article some properties of Markovian mean ergodic operators are studied. As an application of the tools developed, and using the admissibility feature, a “reduction of order” technique for multiparameter admissible superadditive processes is obtained. This technique is later utilized to obtain a.e. convergence of averages $n^{-2}{\sum }_{i,j=0}^{n-1}{f}_{(i,j)}$ as well as their weighted version.

Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^p\left(\nu \right)$, $T₁,...,T_k$, $n̅=(n₁,...,n_k)\in {\mathbb{N}}^k$ and $\alpha ̅=(\alpha ₁,...,{\alpha }_k)$ with $0<{\alpha }_j\le 1$, we define the ergodic Cesàro-α̅ averages ${}_{n̅,\alpha ̅}f=1/\left({\prod }_{j=1}^k{A}_{n_j}^{{\alpha }_j}\right){\sum }_{i_k=0}^{n_k}\cdots {\sum }_{i₁=0}^{n₁}{\prod }_{j=1}^k{A}_{n_j-i_j}^{{\alpha }_j-1}{T}_k^{i_k}\cdots T{₁}^{i₁}f$. For these averages we prove the almost everywhere convergence on X and the convergence in the $L^p\left(\nu \right)$ norm, when $n₁,...,n_k\to \infty$ independently, for all $f\in L^p\left(d\nu \right)$ with p > 1/α⁎ where $\alpha ⁎=mi{n}_{1\le j\le k}{\alpha }_j$. In the limit case p = 1/α⁎, we prove that the averages ${}_{n̅,\alpha ̅}f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $\Lambda (1/\alpha ⁎,{\phi }_{m-1})$ with $\phi ₘ\left(t\right)=t{(1+log⁺t)}^m$. To obtain the result in the limit case we need to study...

Let be a complex, separable Hilbert space of finite or infinite dimension, and let ℬ() be the algebra of all bounded operators on . It is shown that if φ: ℬ() → ℬ() is a multiplicative map(not assumed linear) and if φ is sufficiently close to a linear automorphism of ℬ() in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in ℬ() such that $\phi \left(A\right)=S^{-1}AS$ for all A in ℬ(). When is finite-dimensional, similar results are obtained with the mere assumption that there...

We obtain a sufficient condition on a B(H)-valued function φ for the operator $⨍\mapsto {\Gamma }_{\phi }{⨍}^{\text{'}}\left(S\right)$ to be completely bounded on $H^{\infty }B\left(H\right)$; the Foiaş-Williams-Peller operator | St Γφ | Rφ = | | | 0 S | is then similar to a contraction. We show that if ⨍ : D → B(H) is a bounded analytic function for which $(1-r)\left|\right|{⨍}^{\text{'}}\left(r{e}^{i\theta }\right){\left|\right|}_{B\left(H\right)}^{2}rdrd\theta$ and $(1-r)\left|\right|⨍"\left(r{e}^{i\theta }\right){\left|\right|}_{B\left(H\right)}rdrd\theta$ are Carleson measures, then ⨍ multiplies ${\left(H^1{c}^1\right)}^{\text{'}}$ to itself. Such ⨍ form an algebra A, and when φ’∈ BMO(B(H)), the map $⨍\mapsto {\Gamma }_{\phi }{⨍}^{\text{'}}\left(S\right)$ is bounded $A\to B(H^2\left(H\right),L^2\left(H\right)\ominus H^2\left(H\right))$. Thus we construct a functional calculus for operators of Foiaş-Williams-Peller type.

In this paper, we study multivalued linear operators (MLO's) and their resolvents in non reflexive Banach spaces, introducing a new condition of a minimal growth at infinity, more general than the Hille-Yosida condition. Then we describe the generalized semigroups induced by MLO's. We present a criterion for an MLO to be a generator of a generalized semigroup in an arbitrary Banach space. Finally, we obtain some existence results for differential inclusions with MLO's and various types of multivalued...

Multivariate spectral multipliers for systems of Ornstein-Uhlenbeck operators are studied. We prove that $L^p$-uniform, 1 < p < ∞, spectral multipliers extend to holomorphic functions in some subset of a polysector, depending on p. We also characterize L¹-uniform spectral multipliers and prove a Marcinkiewicz-type multiplier theorem. In the appendix we obtain analogous results for systems of Laguerre operators.