We develop a microspectral theory for quasinilpotent linear operators Q (i.e., those with σ(Q) = {0}) in a Banach space. When such Q is not compact, normal, or nilpotent, the classical spectral theory gives little information, and a somewhat deeper structure can be recovered from microspectral sets in C. Such sets describe, e.g., semigroup generation, resolvent properties, power boundedness as ...

We characterize generators of polynomially bounded C0-semigroups in terms of an integrability condition for the second power of the resolvent on vertical lines. This generalizes results by Gomilko, Shi and Feng on bounded semigroups and by Malejki on polynomially bounded groups.

The paper deals with infinite block matrices having compact off diagonal parts. Bounds for the spectrum are established and estimates for the norm of the resolvent are proposed. Applications to matrix integral operators are also discussed. The main tool is the π-triangular operators defined in the paper.

Let s0 < 0 be the abscissa of absolute convergence of the dynamical zeta function Z(s) for several disjoint strictly convex compact obstacles Ki ⊂ R , i = 1, . . . , κ0, k0 ≥ 3, and let Rχ(z) = χ(−∆D − z2)−1χ, χ ∈ C∞ 0 (R ), be the cut-off resolvent of the Dirichlet Laplacian −∆D in Ω = RN \ ∪0 i=1Ki. We prove that there exists σ1 < s0 such that Z(s) is analytic for Re(s) ≥ σ1 and the cut-off r...

Given a Banach space operator with interior points in the localizable spectrum and without non-trivial divisible subspaces, this article centers around the construction of an infinite-dimensional linear subspace of vectors at which the local resolvent function of the operator is bounded and even admits a continuous extension to the closure of its natural domain. As a consequence, it is shown th...

We study the semiclassical estimates of the resolvent R(+ ii); 2 J R + ; 2]0;1] of a self-adjoint operator L(h) in the space of bounded operators L(H 0;s ; H 0;?s); s > 1=2. In the general case of long-range trapping "black-box" perturbations we prove that the estimate of the cut-oo resolvent k(x)R(+i0)(x)k H!H C exp(Ch ?p); (x) 2 C 1 0 (R n); p 1 implies the estimate kR(+ ii)k s;?s C 1 exp(C 1...

The power (−A)b, b ∈ C is defined for a closed linear operator A whose resolvent is polynomially bounded on the region which is, in general, strictly contained in an acute angle. It is proved that all structural properties of complex powers of densely defined operators with polynomially bounded resolvent remain true in the newly arisen situation. The fractional powers are considered as generato...

We consider a Markov-modulated Brownian motion (MMBM) with phase-dependent termination rates, i.e. while in a phase i the process terminates with a constant hazard rate ri ≥ 0. For such a process we determine the matrix of expected local times (at zero) before termination and hence the resolvent. The results are applied to some recent questions arising in the framework of insurance risk. We fur...