Can we create a proprietary operating system based on the Linux operating system (with GPL license)? If we produce an operating system based on Linux, must we deliver its source code to anyone who wants it?
The Linux copyright is actually many copyrights held by those whose individual contributions to the kernel have been accepted. You would need the permission (i.e., a license) from all these copyright holders to create a derivative work.
The law permits you to copyright a Linux version as a work of your own; however, because of the GNU GPL (General Public License), you would not be allowed to limit the rights of others to use and modify your version of the kernel. The GPL says that in return for being allowed to modify and distribute the work you receive, you must grant the same rights to others, along with the source code you received and contributed.
If you are talking about a Linux operating system (the many services around the kernel that make it an operating system) some of the pieces are licensed under the GPL or LGPL (Lesser General Public License), and others under the BSD license. In the case of the BSD license, you would not need to share the source code to your modifications, but you could not make a complete operating system just out of the parts under the BSD license.
Commercially successful companies that deal in Linux distributions (e.g., Red Hat, SUSE, etc.) cannot depend on copyrights to give them exclusive rights to their version because they must distribute the source code. They depend on reputation, technical proficiency and certifying that the binaries of their distribution will work flawlessly with certain hardware/software. In some cases, the distros grant indemnification against patent suits.
The GPL says that you must deliver or make available the source code to any software licensed under the GPL that you distribute to someone else. If a person has not received the software in question, they have no claim under the GPL to receive the source code. While this might make you believe that you could successfully restrict the source code to your customers, you can't restrict the distribution of GPL'd software; your customers could legally give it away. Hence, there are no proprietary advantages in software under the GPL.