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Fredholm determinants

100%
McKean H.
Open Mathematics
|
2011
|
tom 9
|
nr 2
205-243
EN
The article provides with a down to earth exposition of the Fredholm theory with applications to Brownian motion and KdV equation.
2
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Compact differences of composition operators on weighted Dirichlet spaces

76%
Allen R., Heller K., Pons M.
Open Mathematics
|
2014
|
tom 12
|
nr 7
1040-1051
EN
Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S 2, the space of analytic functions whose first derivative is in H 2, and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.
3
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The Orthogonal Projection and the Riesz Representation Theorem

76%
Narita K., Endou N., Shidama Y.
Formalized Mathematics
|
2015
|
tom 23
|
nr 3
243-252
EN
In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals on real Hilbert spaces. Referring to the article [15], we also defined some definitions on real Hilbert spaces and proved some theorems for defining dual spaces of real Hilbert spaces. As to the properties of all definitions, we proved that they are equivalent properties of functionals on real normed spaces. In Sec. 2, by the definitions [11], we showed properties of the orthogonal complement. Then we proved theorems on the orthogonal decomposition of elements of real Hilbert spaces. They are the last two theorems of existence and uniqueness. In the third and final section, we defined the kernel of linear functionals on real Hilbert spaces. By the last three theorems, we showed the Riesz representation theorem, existence, uniqueness, and the property of the norm of bounded linear functionals on real Hilbert spaces. We referred to [36], [9], [24] and [3] in the formalization.
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