We define the complex powers of a densely defined operator A whose resolvent exists in a suitable region of the complex plane. Generally, this region is strictly contained in an angle and there exists α ∈ [0,∞) such that the resolvent of A is bounded by O((1 + |λ|)α) there. We prove that for some particular choices of a fractional number b, the negative of the fractional power (−A)b is the c.i....

In this paper we examine the Laplacian on the product of two asymptotically hyperbolic spaces from the point of view of geometric scattering theory. In particular, we describe the asymptotic behavior of the resolvent applied to Schwartz functions and that of the resolvent kernel itself. We use these results to find the Martin boundary of the product space. This behaves (nearly) as expected when...

We show that the method of Kipnis and Varadhan (Comm. Math. Phys. 104 (1986) 1) to construct a martingale approximation to an additive functional of a stationary ergodic Markov process via the resolvent is universal in the sense that a martingale approximation exists if and only if the resolvent representation converges. A sufficient condition for the existence of a martingale approximation is ...

Within the framework of generic programming, we implement an abstract algorithm for solution of an integral equation of second kind with the resolvent kernels method. Then, as an application of the idempotent analysis analog of resolvent kernels developed in [LP01], we apply the algorithm to the numerical solution of an optimal control problem with stopping time. 1991 Mathematics Subject Classi...

Within convex analysis, a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago. When one considers the subdifferential of the proximal average, a natural averaging operation of the subdifferentials of the averaged functions emerges. In the present paper we extend the reach of this averaging operation to the...

We discuss magnetic Schrödinger operators perturbed by measures from the generalized Kato class. Using an explicit Krein-like formula for their resolvent, we prove that these operators can be approximated in the strong resolvent sense by magnetic Schrödinger operators with point potentials. Since the spectral problem of the latter operators is solvable, one in fact gets an alternative way to ca...

The resolvent (λI − A)−1 of a matrix A is naturally an analytic function of λ ∈ C, and the eigenvalues are isolated singularities. We compute the Laurent expansion of the resolvent about the eigenvalues of A. Using the Laurent expansion, we prove the Jordan decomposition theorem, prove the Cayley-Hamilton theorem, and determine the minimal polynomial of A. The proofs do not make use of determin...

In this paper, we introduce and consider a new system of extended general variational inclusions involving eight different operators. Using the resolvent operator techniques, we show that the new system of extended general variational inclusions is equivalent to the fixed point problem. We prove the strong convergence of some new explicit iterative parallel algorithms using resolvent methods un...