Very questions on these algebras are motive of modern research. The operators of the Calkin algebra radical are called Reisz operators and can be characterized spectrally and in terms of the dimensions of Rec λI − T k, ker λI − T k ,etc. These are called Fredholm operators and are the operators that give invertible elements of the Calkin algebra. For example, consider the bounded operators in a Banach space with closed range and with kernel and co-kernel of finite dimension.
This theory was created almost immediately after the Fredholm theory, and their beginning is given in the “Lecons de Mécanique Céleste” by Poincaré and Fichot, and to the Hilbert works on contour and boundary problems of the analytic functions theory.Ī possible treatment, bringing the Cauchy ideas together with Banach algebras, is the consideration of the Calkin algebra B X K X ,on a Banach space X ,likewise as the operators of subalgebras of this special Banach algebra (e.g., the algebra of the bounded operators B X ,and K X ,the ideal of compact operators). Likewise, as special case, for their important theory, we can treat the singular integral equations of Cauchy. In which the kernel e − t − sis not of L 2class and gives a continuous spectra, or even, we consider nonlinear integral equations, etc., that represent the last and recent studies on integral equations after of their study considering extensions of the Banach algebras to integral operators that can define to this proposit, for example, to singular integral equations. Ω t − λ ∫ − ∞ + ∞ e − t − s ω s ds = f t, E1 Such is the case, for example, of the Lalesco-Picard integral equation: Likewise, there arise integral equations in which the proper values are corresponded to linearly independent infinite proper functions. Arise numerical and approximate methods on the big vastness that give the Banach algebras, even using probabilistic measures to solve some integral equations in the ambit of distributions and stochastic process. 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, 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.
The themes of recent research are focused on nonlinear integral equations, the new numerical and adaptive methods of resolution of integral equations, the generalization of Fredholm integral equations of second kind, integral equations in time scales and the spectral densities, operator theories for nonsymmetric and symmetric kernels, extension problems to Banach algebras to kernels of integral equations, singular integral equations, special treatments to solve Fredholm integral equations of first and second kinds, nondegenerate kernels and symbols of integral equations, topological methods for the resolution of integral equations and representation problems of operators of integral equations.
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