Fredholm Operators and the Generalized Index
نویسنده
چکیده
One of the most fundamental problems in mathematics is to solve linear equations of the form Tf = g, where T is a linear transformation, g is known, and f is some unknown quantity. The simplest example of this comes from elementary linear algebra, which deals with solutions to matrix-vector equations of the form Ax = b. More generally, if V,W are vector spaces (or, in particular, Hilbert or Banach spaces), we might be interested in solving equations of the form Tv = w where v ∈ V , w ∈W , and T is a linear map from V →W . A more explicit source of examples is that of differential equations. For instance, one can seek solutions to a system of differential equations of the form Ax(t) = x′(t), where, again, A is a matrix. From physics, we get important partial differential equations such as Poisson’s equation: ∆f = g, where ∆ = ∑n i=1 ∂ ∂xi and f and g are functions on R. In general, we can take certain partial differential equations and repackage them in the form Df = g for some suitable differential operator D between Banach spaces.
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