Let T and S be two bounded linear operators from Banach spaces X into Y and suppose that T is Fredholm and dimN(T − λS) is constant in a neighborhood of λ = 0. Let d(T ;S) be the supremum of all r > 0 such that dimN(T − λS) and codim R(T − λS) are constant for all λ with |λ| < r. It is a consequence of more general results due to H. Bart and D.C. Lay (1980) that d(T ;S) = limn→∞ γn(T ;S) , wher...

We consider the C*-algebra, generated by multidimensional integral operators with homogeneous degree $$(-n)$$ kernels and of multiplication radial weak oscillating functions form $$|x|^{i\alpha }$$ . For this algebra, an operator symbolic calculus is constructed. In terms calculus, necessary sufficient conditions for Fredholm property are obtained.

Let L be a bounded linear Fredholm mapping of index zero, mapping a Banach space X into a Banach space 7. Then necessary and sufficient conditions for the solvability of the operator equation Lu = ƒ for ƒ e Y are well known. However the same satisfactory state of affairs does not hold for semilinear operator equations in which a compact nonlinear operator Nu is added to the right hand side of L...

where the domain of integration for the operator is (0,∞). The quantity e(p) does not depend on tn’s. It should be remarked that our tn corresponds to tn/2 of refs.[1, 2]. Independently, Bernard and LeClair showed that φ(t) solves the shG equation[3]. Quite recently, Tracy and Widom proved that above φ(t) satisfies both the mKdV hierarchy and the shG hierarchy[2]. Their proof is based on the fa...

In this paper, we consider the transmission eigenvalue problem for Maxwell’s equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We formulate the transmission eigenvalue problem as an equivalent homogeneous system of the boundary integral equation and, assuming that the contrast is constant near the boundary of the...

Let T and S be two bounded linear operators from Banach spaces X into Y and suppose that T is Fredholm and the stability number k(T ;S) is 0. Let d(T ;S) be the supremum of all r > 0 such that dimN(T − λS) and codim R(T − λS) are constant for all λ with |λ| < r. It was proved in 1980 by H. Bart and D.C. Lay that d(T ;S) = limn→∞ γn(T ;S) , where γn(T ;S) are some non-negative (extended) real nu...