This　paper　gives　the　concepts　of　finite　dimensional　irreducible　operators　((FDI)　operators)　and　infinite　dimensional　irreducible　operators　((IDI)　operators).　Discusses　the　relationships　of　(FDI)　operators,　(IDI)　operators　and　strongly　irreducible　operators　((SI)　operators)　and　illustrates　some　properties　of　the　three　classes　of　operators.　Some　sufficient　conditions　for　the　finite-dimensional　irreducibility　of　operators　which　have　the　forms　of　upper　triangular　operator　matrices　are　given.　This　paper　proves　that　every　operator　with　a　singleton　spectrum　is　a　small　compact　perturbation　of　an　(FDI)　operator　on　separable　Banach　spaces　and　shows　that　every　bounded　linear　operator　T　can　be　approximated　by　operators　in　(ΣFDI)(X)　with　respect　to　the　strong-operator　topology　and　every　compact　operator　K　can　be　approximated　by　operators　in　(ΣFDI)(X)　with　respect　to　the　norm　topology　on　a　Banach　space　X　with　a　Schauder　basis,　where　(ΣFDI)(X):=　{T　∈　B(X):　T　=　Σi=1k　⊕Ti,　Ti　∈　(FDI),　k　∈　ℕ}.　©　2015,　Institute　of　Mathematics,　Academy　of　Mathematics　and　Systems　Science,　Chinese　Academy　of　Sciences,　Chinese　Mathematical　Society　and　Springer-Verlag　Berlin　Heidelberg.