We consider the operator $-A+iB$ on a complex Hilbert space, where $A$ is positive self-adjoint and $B$ is self-adjoint, and where, moreover, «$B$ is comparable to $A^{\alpha }$, $\alpha \ge 1$», in a technical sense. Two applications are given.

We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form ẋ = A(t)x + f(t) (*), with f having precompact range, which is then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonant case where $\overline{e^{isp\left(f\right)}}$ may intersect the spectrum of the monodromy operator P of (*) (here sp(f) denotes the Carleman spectrum of f). We show that if (*) has a bounded...

We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $dU\left(t\right)=A\left(t\right)U\left(t\right)dt+B\left(t\right)d{W}_H\left(t\right)$, t ∈ [0,T], U(0) = u₀. Here, $W_H$ is a cylindrical Brownian motion on a real separable Hilbert space H, ${\left(B\left(t\right)\right)}_{t\in [0,T]}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, ${\left(A\left(t\right)\right)}_{t\in [0,T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting of evolution...

We show that a positive semigroup $T_t$ on $L_p(\Omega ,\nu )$ with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the $L_p$-scale, which may be of independent interest.

In this paper, we investigate a class of abstract degenerate fractional differential equations with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.

We study existence, uniqueness and form of solutions to the equation $\alpha g-\beta g^{\text{'}}+\gamma g_e=f$ where α, β, γ and f are given, and $g_e$ stands for the even part of a searched-for differentiable function g. This equation emerged naturally as a result of the analysis of the distribution of a certain random process modelling a population genetics phenomenon.

Let $T=T\left(u\right):u\in {ℝ}_d^{+}$ be a strongly continuous d-dimensional semigroup of linear contractions on $L_1((\Omega ,\Sigma ,\mu );X)$, where (Ω,Σ,μ) is a σ-finite measure space and X is a reflexive Banach space. Since $L_1((\Omega ,\Sigma ,\mu );X)*=L_{\infty }((\Omega ,\Sigma ,\mu );X*)$, the adjoint semigroup $T*=T*\left(u\right):u\in {ℝ}_d^{+}$ becomes a weak*-continuous semigroup of linear contractions acting on $L_{\infty }((\Omega ,\Sigma ,\mu );X*)$. In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), $u\in {ℝ}_d^{+}$, has a contraction majorant P(u) defined on $L_1((\Omega ,\Sigma ,\mu );ℝ)$, that is, P(u) is a positive linear contraction on $L_1((\Omega ,\Sigma ,\mu );ℝ)$ such that $\Vert T\left(u\right)f\left(\omega \right)\Vert \le P\left(u\right)\Vert f\left(·\right)\Vert \left(\omega \right)$ almost everywhere...