Re: bijective operators
From: Pouya D. Tafti (p.d.tafti_at_ieee.org)
Date: 01/24/05
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Date: Mon, 24 Jan 2005 01:30:05 +0000 (UTC)
David C. Ullrich wrote:
> Sorry, but this doesn't seem to make much sense. There are no
> conditions on U, V, X, Y, V which will make r invertible - that
> depends on r.
>
> Maybe you have some specific r in mind - if so you need to tell
> us what it is. (What you _wrote_ above the *********** doesn't
> seem to have any bearing on the question - probably there is
> a connection you had in mind, but if so it's because of
> something you haven't told us.)
> ************************
Thanks Dr. Ullrich. Obviously I am confused and have to think more. So
please pardon me and kindly disregard this post if it doesn't make any
more sense.
I do have some specific r in mind, and the reason I was looking for
conditions on X,Y,V was that the definition of r depends on them [1].
However, for now I'll be focusing on conditions on r itself, and will
try to translate these conditions to conditions on X,Y,V at a later
stage.
Let {x_i},{y_i},{v_i} be given bases for X,Y,Z.
We have U=X x Y and r:U' -> V'. Let
v' = r(u')
for some u=(x,y) \in U.
For the case where X,Y,V are finite-dimensional, v'=r(x',y') is
determined by indicating its effect on {v_i}:
(ru'){v_i} = v'{v_i} := W x'{x_i} + S y'{y_i}.
Here v'{v_i} should be interpreted as the (column) vector composed of
{v' v_i} (similarly for x'{x_i},y'{y_i}. I don't know the standard
notation.) W,S are matrices.
In this case the invertibility of r is equivalent to the invertibility
of the matrix [W S].
Now my question is: How can I state a similar formulation for the case
where X,Y,V are infinite-dimensional and a basis for each of them is
given?
I would also greatly appreciate any advice regarding this problem and
recommendations on references. Apologies again if this thread was not
appropriate for s.m.r.
Thanks,
Pouya
[1] The dependence of r on X,Y,Z is as follows. Each element of {v_i}
is associated with a subset M_i of {x_i} and a subset N_i of {y_i}.
rv_i is then defined to be equal to <P,v_i> for a certain P \in H that
interpolates the constraints given in the form
<P,x_i>=m_i for x_i \in M_i,
<P,y_i>=n_i for y_i \in N_i.
I did not mention this earlier because I thought it would complicate
the question beyond my control.
-- Pouya D. Tafti http://grads.ece.mcmaster.ca/~pouya p dot d dot tafti at ieee dot org
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