You own the copyright in your work from the time it is fixed in a tangible medium. There is nothing more you have to do.
You can, however, for a fee, register your copyright with the Copyright Office in the Library of Congress if you so choose. Registering it gives you the ability to sue for infringement and recover statutory damages beyond actual damages.
Copyright is actually a bundle of rights. These include the right to reproduce the work, to create derivative works (such as a screenplay from a novel), to distribute copies, to perform the work, and to display the work. What UMI/ProQuest and the University are asking for is the non-exclusive right to distribute your work. You still own the copyright. You still control everything the statute grants you.
If you are submitting your thesis to UMI/ProQuest, they will list your thesis in their product and sell microfilm or print copies to anyone who asks. You would get a small royalty from the sale. If you sign the Distribution and Reproduction Rights form from the library, we will post our copy in an open access repository and include it in the Networked Digital Library of Theses and Dissertations under the assumption that authors want their work to be read.
Even though the thesis is in the open access repository, you still control the copyright and could grant anyone else the non-exclusive right to distribute the work or the exclusive right to reproduce the work, etc.
If you have further questions, see this useful link or feel free to contact:
Head, Research & Learning / Interim Senior Information Resources Officer
University of Arizona Libraries