Abstract: In this paper, we show that the invertible operator T, which is a bounded linear functional on a separable Hilbert space H, could factor as T=US, where U is unitary and 5 belongs to width-two CSL algebra alg when nest M or N is a countable nest, or S belongs to alg when nests M and N are countable nests. For the factorization of nest,we obtain that T factors as T=US where S∈D^-1_N and U is unitary as N be a countable nest.

Key words: width-two CSL algebra, nest algebra, factorization, invertible operator