In L2(ℝd; ℂn), we consider a wide class of matrix elliptic second order differential operators ε with rapidly oscillating coefficients (depending on x/ε). For a fixed τ > 0 and small ε > 0, we find approximation of the operator exponential exp(− ετ) in the (L2(ℝd; ℂn) → H1(ℝd; ℂn))-operator norm with an error term of order ε. In this approximation, the corrector is taken...
Our aim in this paper is to prove that every separable infinite-dimensional complex Banach space admits a topologically mixing holomorphic uniformly continuous semigroup and to characterize the mixing property for semigroups of operators. A concrete characterization of being topologically mixing for the translation semigroup on weighted spaces of functions is also given. Moreover, we prove that there exists a commutative algebra of operators containing both a chaotic operator and an operator which...
Hypercyclicity of C0-semigroups is a very unstable property: We give examples to show that adding arbitrary small constants or a bounded rank one operator to the generator of a hypercyclic semigroup can destroy hypercyclicity. Also the limit of hypercyclic semigroups (even in operator norm topology) need not be hypercyclic, and a hypercyclic semigroup can be the limit of nonhypercyclic ones. Hypercyclicity is not inherited by the Yosida approximations. Finally, the restriction of a hypercyclic...