Decompositions

Decomposition means separation into “parts”. As far as operators are concerned this usually is done by product (factorization) or by sum. For instance, the polar decomposition says that every operator can be factorized as the product of a partial isometry and a nonnegative operator. On the other hand, the Cartesian decomposition is one by (ordinary) sum: every operator T can be written as T = Re(T) + i Im(T) where Re(T) = 1/2(T + T*) and Im(T) = -i/2(T - T* are self-adjoint operators. We shall however not deal with factorization and ordinary sum decomposition here because, in spite of being useful tools in operator theory, they miss a crucial feature: they do not transfer invariant subspaces from the “parts” (factors or ordinary summands) to the original (decomposed) operator. For the lack of a better name let us say that they do not “isolate the parts”. We shall deal with decomposition by direct sums instead, which do “isolate the parts”: an invariant subspace for a direct summand is invariant for the direct sum.