Re: Questions on pseudoinverses

From: Robert Israel (israel_at_math.ubc.ca)
Date: 03/18/05 

Date: 18 Mar 2005 14:21:08 -0500

In article <d1er9k$1ce$1@dizzy.math.ohio-state.edu>,
Mario S. Mommer <m_mommer@yahoo.com> wrote:

>Let V,W be separable Hilbert spaces, and let L_i:V->W, with i in |N,
>be a sequence of bounded operators with closed range that converge
>pointwise to an operator L:V->W which is also bounded, and which also
>has closed range. All of these operators may be singular.

>Now, suppose that the pseudoinverses of the L_i are all uniformly
>bounded, i.e.

> || L_i^\dagger || < C < infinity for all i in |N.

>Does it follow that the L_i^\dagger converge pointwise?

No. Consider the case V = W = \ell^2, the square-integrable sequences.
Let L be the left shift operator (Lx)_j = x_{j+1}, and L_i = L^i.
These converge pointwise, i.e. strongly, to the operator 0, and
are surjective so they have closed range. The pseudoinverse
L_i^\dagger = R^i where R is the right shift, and has norm 1.
But the R^i do not converge strongly (although they do converge
weakly to 0).

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada