• Dr. Ranjan Kumar Singh

In this paper, we prove two interesting cases of the Fredholm alternative on Banach spaces, one is that for any compact operator T & identity operator I on a Banach spaces X with a non-zero complex number , the injectivity & surjectivity of (T- I) are equivalent and the other one is that either (T- I) is bijective or has non-trivial Kernel and nontrivial Co-Kernel of the same dimension.

Fredholm alternative, compact operator, Banach space, Kernel &Co Kernel.

"THE FREDHOLM ALTERNATIVE AND COMPACT OPERATORS ON BANACH SPACES", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.7, Issue 2, page no.151-153, February-2020, Available :http://www.jetir.org/papers/JETIR2002223.pdf

"THE FREDHOLM ALTERNATIVE AND COMPACT OPERATORS ON BANACH SPACES", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.7, Issue 2, page no. pp151-153, February-2020, Available at : http://www.jetir.org/papers/JETIR2002223.pdf