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In this note we obtain some characterizations of Banach spaces which are not isomorphic to any of their proper subspaces. In particular, we show that a Banach space X is not isomorphic to any of its proper subspaces if and only the equality σ RD (LT) = σ LD (T) holds for every bounded linear operator T on X if and only if int(σ(T)) ⊆ σ LD (T) holds for every bounded linear operator T on X, where L T  denotes the left multiplication operator by T.

Characterizations of Banach Spaces which are not Isomorphic to any of their Proper Subspaces