Narrow Results

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

This paper addresses the existence of solutions u ∈ H¹(ℝ₊;ℝ^N) of ODE systems , with boundary condition u₁(0) = ξ, where u₁ is a (vector) component of u. Under general conditions, the problem corresponds to a functional equation involving a Fredholm operator with calculable index, which is proper on the closed bounded subsets of H¹(ℝ₊;ℝ^N). When the index is 0 and the solutions are bounded a priori, the existence follows from an available degree theory for such operators. Specific conditions are given that guarantee the existence of a priori bounds and second order equations with Dirichlet, Neumann or initial value conditions are discussed as applications.

The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single expression the Atiyah–Singer formula for elliptic operators, as well as Boutet de Monvel's Toeplitz index formula.

In this article we show that holomorphic Fredholm-valued mappings defined on connected pseudo-convex domains in Banach spaces with unconditional basis always have meromorphic generalised inverses. We show they have holomorphic generalised inverses if and only if the kernels have the same dimension at all points in Ω.

The spectral picture, the question of hyponormality, and other spectral decomposition properties are discussed for the class of generalized Cesàro operators with rational symbol on Hardy and Bergman spaces.