Let 0 < p ≤ q ≤ +∞. Let T be a bounded sublinear operator from a Banach space X into an Lp(Ω,μ) and let ∇T be the set of all linear operators ≤ T . In the present paper, we will show the following. Let C be a positive constant. For all u in ∇T , Cpq(u) ≤ C (i.e., u admits a factorization of the form X ũ → Lq(Ω,μ) Mgu → Lp(Ω,μ), where ũ is a bounded linear operator with ‖ũ‖ ≤ C , Mgu is the bo...

A bounded linear operator T on a complex Hilbert space H is irreducible if it has no reducing subspace other than the trivial ones (0) and H; it is strongly irreducible if every operator similar to T is irreducible. Equivalently, T is irreducible (resp. strongly irreducible) if the only projections (resp. idempotent operators) commuting with T are 0 and I. Since (strongly) irreducible operators...

It easy to see that every bounded linear operator is continuous. It is not hard to show that the converse is also true. The notions of bounded and continuous thereby coincide for linear operators acting between normed spaces. It is customary to prefer the terminology bounded linear operator over that of continuous linear operator. The reason for this preference is the fact that the hard part of...

Given a Hilbert space and the generator A of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any g(−s) ∈ H∞ we show that there exists an inf nite-time admissible output operator g(A). If g is rational, then this operator is bounded, and equals the “normal” def nition of g(A). In particular, when g(s) = 1/(s + α), α ∈ C + 0 , then this admissible output operator...

Let X be a real separable Hilbert space. Q linear, bounded, positive and compact operator on let A : Dom ( ) ⊆ → self-adjoint generating strongly continuous semigroup . F (smooth enough) function { W t } ≥ 0 -valued cylindrical Wiener process. For any α , we are interested in the mild solution x of semilinear stochastic partial differential equation d = + > ; ∈ its associated transition (0.1) P...

Let H be a real Hilbert space. Consider on H a nonexpansive family T = {T(t) : 0 ≤ t < ∞} with a common fixed point, a contraction f with the coefficient 0 0. Assume that 0 < γ < γ̄ /α and that S = {S (t) : 0 ≤ t < ∞} is a family of nonexpansive self-mappings on H such that F(T ) ⊆ F(S) and T has prop...