I need some books to learn the basis of linear operator theory and the spectral theory with, if it's possible, physics application to quantum mechanics. Can somebody help me?
I think a good, and classic, reference for your case is the following,
The very last chapter of Kreyszig deals with Quantum Mechanics.
And, once you've learned how to "translate" the language of Functional Analysis into that of Quantum Mechanics, you can go to more advanced texts in specific topics.
There are literally hundreds of introductory books on linear operators and their use in quantum mechanics. Finding the right one for you can be tricky. Let me recommend two that have been successful enough to be republished as Dover paperbacks. They're both by Thomas F. Jordan and are specifically oriented towards quantum mechanics applications. The most elementary one (perhaps too elementary) is,
"Quantum Mechanics in Simple Matrix Form" (1986) Dover (2005) ISBN 0-486-44530-5
The other one, significantly more sophisticated, but still very accessible is,
"Linear Operators for Quantum Mechanics" (1997) Dover (2006) ISBN 0-486-45329-4
For a fast and useful introduction, you can read the second chapter of the book
"Quantum Mechanics, Vol. I", by Claude Cohen-Tannoudji et al.
You will find here almost all you need to study quantum physics.
For a deeper treatment, you can try,
"Functional Analysis. Volume I.", by Reed & Simon. This is the first volume of the collection Methods of Modern Mathematical Physics.