A comprehensive study on the invariant subspace problem for various operators on banach spaces

In this thesis, the aim is to conduct a comprehensive study presenting both positive and negative results related to the Invariant Subspace Problem for various bounded linear operators on Banach spaces. To achieve this, the problem is examined separately in Hilbert spaces too, in accordance with the existing literature. Operators are classified according to the spaces on which they act. The study begins by considering finite-dimensional and non-separable spaces, presenting positive results in the existing literature. Subsequently, various classes of operators are analyzed, progressing from general to more specific cases, with corresponding positive results provided. In the next stage, Per Enflo's modern approach to the existence of non-trivial invariant subspaces for injective operators on Hilbert spaces is briefly examined, and relevant results are discussed. Finally, new techniques are proposed, based on analyzing the rotation and norm behavior of vectors under bounded linear operators. These methods aim either to control the rotational action of operators or to identify subspaces on which a given operator $T$ behaves as a scalar multiple of an isometry. This approach not only offers a potential framework for identifying non-trivial invariant subspaces for $T$ but also enables the construction of new bounded linear operators on larger Banach spaces by extending operators that already possess non-trivial invariant subspaces.