Direct sum of integration operators (Q1073319)

Let \({\mathfrak h}_ j\) \((j=1,2)\) be subspaces in \({\mathfrak h}\) and \(A_ j\) be bounded operators in \({\mathfrak h}\). If either \(h_ 1\perp {\mathfrak h}_ 2\) or \({\mathfrak h}_ 1\dot +{\mathfrak h}_ 2\) is the uniform direct sum of one of the subspaces and while \(A_ 2=U_{12}*A_ 1U_{12}\), where \(U_{12}\) is the natural unitary mapping of the space \({\mathfrak h}_ 2\) onto \({\mathfrak h}_ 1\) then the block (\({\mathfrak h}_ 1,{\mathfrak h}_ 2,A_ 1,A_ 2)\) is called to be basic. In this paper the author proves: Let \({\mathfrak h}={\mathfrak h}_ 1\dot +...\dot +{\mathfrak h}_ n\) and assume that in \({\mathfrak h}_ j\) \((j=1,...,n)\) is given the opertor \(A_ j\in \Lambda_ 1^{\ell}\), the set of all completely non-self-adjoint Volterra dissipative operators with n-dimensional imaginary component \(A_ I\) and sp \(A_ I=\ell\). The linear operator \(Ah=A_ jh\) (h\(\in {\mathfrak h}_ j)\) acting in \({\mathfrak h}\) is a completely non-self-adjoint Volterra dissipative operator with nuclear imaginary component if and only if each block (\({\mathfrak h}_ i,{\mathfrak h}_ j,A_ i,A_ j)\) is basic.