On a factorization of operators as a product of an essentially unitary operator and a strongly irreducible operator

Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem

“…We answer this question for quaternionic normal operators by proving a factorization in a strongly irreducible sense, by means of decomposing the given operator as a product of a sufficiently small compact perturbation of a partial isometry and a strongly irreducible quaternionic operator. Our result is the quaternionic version (for normal operators) of the result proved recently in [15] by G. Tian et al In which, the authors employed the properties of Cowen-Douglas operators related to complex geometry on complex Hilbert spaces. Also see [12] for such factorization in finite dimensional Hilbert spaces.…”

“…Later, we prove a factorization of quaternionic normal operator in a strongly irreducible sense, by means of replacing the partial isometry W by a desirably small compact perturbation of W and |T | is replaced by strongly irreducible operator. It is a quaternionic extension (for normal operators) of the result proved in [15].…”

On a factorization of operators on finite dimensional Hilbert spaces

“…The proof of the theorem is quite different from the proofs in [8]. We will give it in the next section.…”

“…A natural question can be asked: can we write an operator into a product of a partial isometry and an operator having fewer reducible subspaces? In reference [8], the authors answered the question when H is complex separable infinite dimensional. More precisely, on a complex separable infinite dimensional Hilbert space, for any operator T and any ε > 0, there exists a decomposition T = (U + K)S , where U is a partial isometry, K is a compact operator with ||K|| < ε, and S is strongly irreducible.…”