On the Fredholm Theory of Integral Equations for Operators Belonging to the Trace Class of a General Banach Space

Introductory Chapter: Frontier Research on Integral Equations and Recent Results

“…The themes of recent research are focused on nonlinear integral equations [1], the new numerical and adaptive methods of resolution of integral equations [2], the generalization of Fredholm integral equations [3] of second kind, integral equations in time scales and the spectral densities [3,4], operator theories for nonsymmetric and symmetric kernels [1,5], extension problems to Banach algebras to kernels of integral equations [5][6][7], singular integral equations [10], special treatments to solve Fredholm integral equations of first and second kinds, nondegenerate kernels [3,6] and symbols of integral equations [7], topological methods for the resolution of integral equations and representation problems of operators of integral equations. Now, well, the field of the integral equations is not finished yet, not much less with the integral equations for which the Fredholm theorem is worth [fredholm], nor with the completely continuous operators, since there exist other integral equations developed of the Hilbert theory respect to the Fredholm discussion, and studies on singular integral equations, also by Hilbert, Wiener and others [8].…”

The flat-trace asymptotics of a uniform system of contractions

“…In [Rus,p. 117], Ruston deduces (from Hadamard's inequality, following [Sm, p. 120]) log | det(/ -Y)\ < \x{Y) 2 -Re(Tr (K)) for any finite rank operator Y on a Banach space.…”

An application of Plemelj-Smithies formulas to computing generalized inverses of Fredholm operators

“…The theory has been extended by many authors, first by Riesz, Hilbert and Carleman, to square-integrable kernels. Grothendieck [2] and Ruston [3] have generalized the theory to nuclear operators in Banach spaces, Leżański [4], Sikorski [5][6][7] and Buraczewski [8,9] to quasinuclear operators in Banach spaces. Later papers of Pietsch [10,11] and nig o K ɺ ɺ [12] dealt with absolutely 2-summing operators and absolutely p-summing operators ( 2 > p ) in Banach spaces, respectively.…”