What does a rate in a mean ergodic theorem imply?
We develop a general framework for the inverse mean ergodic theorems with rates for operator semigroups thus completing a construction of the theory initiated in [16] and [17].
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arXiv subjects
Alexander Gomilko
Publications and source records attributed to Alexander Gomilko.
On convergence rates in approximation theory for operator semigroups
We create a new, functional calculus, approach to approximation of C_0-semigroups on Banach spaces. As an application of this approach, we obtain optimal convergence rates in classical approximation formulas for C_0-semigroups. In fact, our methods allow one to derive a number of similar formulas and equip them with sharp convergence rates. As a byproduct, we prove a new interpolation principle leading to efficient norm estimates in the Banach algebra of Laplace transforms of bounded measures on the semi-axis.
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On rates in Euler's formula for C_0-semigroups
By functional calculus methods, we obtain optimal convergence rates in Euler's approximation formula for C_0-semigroups restricted to ranges of generalized Stieltjes functions. Our results include a number of partial cases studied in the literature and cannot essentially be improved.
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Bernstein functions and rates in mean ergodic theorems for operator semigroups
We present a functional calculus approach to the study of rates of decay in mean ergodic theorems for bounded strongly continuous operator semigroups. A central role is played by operators of the form $g(A)$, where $-A$ is the generator of the semigroup and $g$ is a Bernstein function. In addition, we obtain some new results on Bernstein functions that are of independent interest.
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On a Question of A. E. Nussbaum on Measurability of Families of Closed Linear Operators in a Hilbert Space
The purpose of this note is to answer a question A. E. Nussbaum formulated in 1964 about the possible equivalence between weak measurability of a family of densely defined, closed operators T(t), t real, in a separable complex Hilbert space H on one hand, and the notion of measurability of the 2 \times 2 operator-valued matrix of projections onto the graph Gamma(T(t)) of T(t) on the other, in the negative. Our results demonstrate an interesting distinction between the direct integral over the family of operators T(t) with respect to Lebesgue measure and the naturally maximally defined operator associated with pointwise application of T(t) in the vector-valued Hilbert space L^2(\mathbb R; dt; H). We also provide explicit criteria for the measurability of the matrix of projections onto the graph Gamma(T(t)) of T(t).
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