In analyzes functional, a operator with core is a standard example of operator. He is defined on functional spaces on which integration has a direction, and by means of product of convolution by functions depending on parameters. The family of functions by which the convolution takes place call the core of the operator. An operator with core thus puts himself in the form:

$S\; (F)\; (T)\; =\; \backslash \; int\_Af\; (X).\; K\; (X,\; T)\; of\; (T)\; the$, without more precision on the context, where T is a parameter, F is the function to which the operator is applied and K the core of the operator.

According to the situations, fields of definition of the concerned functions, and the regularity of the functions intervening in the problems, functional spaces can be modified. However, in a great number of already interesting cases in practice, there exists a complete study of the spectral analysis of the operators with cores. In addition, it is frequent that the operators with cores are compact operators, in other words send parts limited on relatively compact parts.

The operators with cores intervene in the phenomena of diffusion where intervene classically of the integral equations. The existence and the unicity of the solutions find solutions with the Alternative of Frédholm, when the latter is applicable, IE when the operator with core is a compact operator.

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