The Weiss Conjecture for Bounded Analytic Semigroups

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

“…We further note that the estimate (2.10) is also used for the Weiss conjecture in control theory (cf. [27] and references therein).…”

On square functions associated to sectorial operators

“…In [16], we studied the case when T t = e −tA is a bounded analytic semigroup, that is, there exists α > 0 such that (T t ) t>0 extends to a bounded analytic family (e −zA ) z∈Σα ⊂ B(X). We proved the following result (see [16,Theorem 4.1]). Recall that T t = e −tA is a bounded analytic semigroup on X if and only if A satisfies the conditions (S1) and (S2) from Section 1 for some ω < 1 2 π. Define…”

“…This new concept is a natural extension of the classical notion of admissibility considered e.g. in [24], [23], [25], [8] or [16]. Given a bounded analytic semigroup T t = e −tA on L p (Ω) and a linear mapping C from the domain of A into some L q (Σ), we will study conditions under which we have an estimate of the form…”

Square functions, bounded analytic semigroups, and applications

“…(see [34] for details). Applying (4.7), we deduce that for some constant D ′′ not depending on t > 0, we have…”

“…(2) We say that A satisfies a square function estimate if for some (equivalently, for all) F ∈ H ∞ 0 (Σ ω+ ), there is a constant K > 0 such that x F ≤ K x for any x ∈ H. By Theorem 3.2 (and Remark 2.6), A has a bounded H ∞ functional calculus if and only if both A and A * satisfy a square function estimate. In [34], an example is given of an operator A which satisfies a square function estimate, although it does not have a bounded H ∞ functional calculus. (3) Assume that A is 1-1, and let F ∈ H ∞ 0 (Σ ω+ ) \ {0}.…”