On the unique finite (SI) decomposition of the Jordan operator \(\bigoplus_{i = 1}^n S(\theta)\) for certain inner function \(\theta\) (Q1566824)

An operator \(T\) in \(L(H)\), where \(L(H)\) denotes the algebra of all bounded operators on a separable Hilbert space \(H\), is said to be strongly irreducible (denoted by \(T\in (\text{SI})\)) if every idempotent in \(L(H)\) commuting with \(T\) is \(0\) or \(I_h\). A finite set of idempotents \(\{P_i\}^n_{i= 1}\) in \(L(H)\) is said to be a decomposable unit of \(T\), if (i) \(P_i\in \{T\}'\) -- the commutant of \(T\) and \(P_iP_j= 0\), \(i\neq j\), \(1\leq i,j\leq n\) and (ii) \(\sum^n_{i=1} P_i= I_H\) and \(T|_{P_iH}\in (\text{SI})\), \(i= 1,2,\dots, n\). An operator \(T\) in \(L(H)\) is said to have finite (SI) decomposition if for every non-zero idempotent \(P\in \{T\}'\), \(T|_{pH}\) has a decomposable unit. Let \(T\in L(H)\) have finite (SI) decomposition. We say that \(T\) has unique finite (SI) decomposition upto similarity if for every two decomposable units of \(T\), \(\{P_i\}^n_{i= 1}\) and \(\{Q_i\}^m_{i=1}\), we have that \(n= m\) and there is an invertible operator \(X\) in \(\{T\}'\) such that \(\{XP_iX^{- 1}\}^n_{i= 1}= \{Q_i\}^n_{i= 1}\). \(H^\infty\) denotes the Banach algebra of all bounded holomorphic functions \(f(z)\) on the open unit disc \(D\) with the norm \(\|f\|= \sup_{z\in D}|f(z)|\). Let \(S\) be the unilateral shift on the Hardy space \(H^2\) given by \(Sf(z)= zf(z)\), \(f\in H^2\). Let \(\Theta\in H^\infty\) be a nonconstant inner function and let \(P_\Theta\) be the orthogonal projection of \(H^2\) onto \(H(\Theta)= H^2\ominus\Theta H^2\). The Jordan block \(S(\Theta)\) is defined by \(s(\Theta)= P_\Theta S|_{H(\Theta)}\). The authors show that for the singular function \(\Theta\), \(S(\Theta)\in (\text{SI})\), \(\bigoplus^n_{i=1} S(\Theta)\) has unique finite (SI) decomposition upto similarity. Applying these results, it is shown that \(\bigoplus^n_{i= 1}V\), where \(V\) denotes the Volterra operator on \(L^2([0, 1])\) has unique finite (SI) decomposition upto a similarity.