(no subject)
So today I got through quite an elegent result in math: Beurling's theorem. It completely characterizes the invariant subspace of the unilaterial shift on l_2. The idea is to switch from l_2 to H^2 (the Hardy space) and so using Hardy space theory, these subspaces are completely determined by (and to each combination one can associate an invariant subspace to): a sequence of complex numbers a_n where lim |a_n| = 1 and sum (1-|a_n|) converges, and a finite Borel measure on the circle that's singular with respect to Lebesgue measure.
Nice thing is that it's very conceptual and not hard at all (I'm using the complex analysis part of 'Adult Rudin.')
The thing is though, I have to wonder: why should I, or anyone for that matter, even care about invariant subspaces of the shift? Is the shift operator at all useful? Or do people do this stuff for the hell of it? Who knows...
Nice thing is that it's very conceptual and not hard at all (I'm using the complex analysis part of 'Adult Rudin.')
The thing is though, I have to wonder: why should I, or anyone for that matter, even care about invariant subspaces of the shift? Is the shift operator at all useful? Or do people do this stuff for the hell of it? Who knows...