Criteria for the similarity of a dissipative integral operator to a normal operator (Q2783038)

Let \(H\) be a separable Hilbert space, \(\mu\) a finite positive measure on \([0,1]\). Let \(\alpha\) be a function from \([0,1]\) to the space of bounded linear operators \(L(H)\), \(k(x,s)\) an \(L(H)\)-valued positive definite kernel with \(\operatorname {tr}k(x,x)\in L^1(\mu)\). Suppose that the \(\alpha(x)\) are selfadjoint for all \(x\in [0,1]\). Define the operator \(A\) on \(L^2(H,\mu)\) by the formula NEWLINE\[NEWLINE(Af)(x)=\alpha(x)f(x)+\tfrac 12 i\mu(\{x\})k(x,x)f(x)+i\int_{[0,x)}k(x,s)f(s) d\mu(s).NEWLINE\]NEWLINE The authors of the paper under review study conditions ensuring that the operator \(A\) is similar to a normal operator.