A classical problem in operator theory has been to determine the spectrum of Toeplitz-like operators on Hilbert spaces of vector-valued holomorphic functions on the open unit ball in Cm. In this note we obtain necessary conditions for k-tuples of such operators to be Fredholm in the sense of Taylor and show they are sufficient in the case of the unit disk.

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators in L2(Ω; dnx), where Ω ⊂ Rn, n = 2, 3, are open sets with a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in L2(Ω; dnx) to modifie...

The operator Lry = / + Py, whose domain is determined in part by the Stieltjes integral boundary condition Jo dv{i)y{f) = 0, is studied in Xj¡($>, 1), 1 < p < oo. It is shown that Lp has a dense domain; hence there exists a dual operator L* operating on .£¡¡(0,1). After finding LJ we show that both L, and L¡¡ are Fredholm operators. This implies some elementary results concerning the spectrum a...

We introduce generalized operator valued Jost solutions of first order ill-posed differential equations on Hilbert spaces. We then construct an infinite dimensional Evans function for abstract differential equations as a 2– modified Fredholm determinant of the operator obtained by adding the values at zero of the generalized operator valued Jost solutions. Next, we prove a formula that connects...

This article investigates an equation with a rapidly oscillating inhomogeneity and decreasing kernel of integral operator Fredholm type. Earlier, differential problems this type were studied in which the term was either absent or had form Volterra-type integral. The presence its significantly affect development algorithm for asymptotic solutions, implementation it is necessary to take into acco...

The quantized Hall conductance in a plateau is related to the index of a Fredholm operator. In this paper we describe the generic “phase diagram” of Fredholm indices associated with bounded and Toeplitz operators. We discuss the possible relevance of our results to the phase diagram of disordered integer quantum Hall systems. The Hall conductance of Integer Quantum Hall systems is described mat...