Schauder's Fixed Point Theorem and the Invariant Subspace Problem*

One of the most outstanding, difficult and long open problem in operator theory is the so-called ‘Invariant Subspace Problem’ which asks the question: Does every bounded linear operator on an infinite-dimensional separable complex Hilbert space have a nontrivial invariant subspace? The problem which is so easy to state is, in fact, very complex in nature. Although it has drawn the attention of many mighty mathematicians and a lot of research work has been done over the decades on its ramifications, no solution of the problem so far seems to be in sight [48], [51].