Click on the links below to see some pictures of the level sets

for some values of the parameters

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

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July 1, 2010 at 00:22 |

Here is an attempt to apply dilation theory in conjecture 9. The idea is that any contraction on the Hilbert space has a unitary power dilation, that is, is a subspace of a larger Hilbert space and there is a unitary operator with the nice property that for all , where represents the orthogonal projection. This suggests the following modification of conjecture 9:

Claim AThere exists and there are unit vectors such that,

for each ,

where denotes the orthogonal projection.

Now, consider the vectors , and notice that

and therefore

for all I wonder if this kind of argument can be turned into a proof that FISP2 is a consequence of claim A. Notice that the spectral theorem can be applied to represent the unitary operator on the Hilbert space as a multiplication operator, that is, where is a measurable function and a.e.

It is clear that claim A has the following equivalent formulation:

Claim BThere exists and there is a function such that,

for each

July 1, 2010 at 00:22 |

Claim.If is a unitary operator on then for every there are unit vectors such that,

July 1, 2010 at 00:23 |

Claim.If is a unitary operator on then for every there are unit vectors such thatJuly 1, 2010 at 00:25 |

Claim.If is a unitary operator on then for every there are unit vectors such that