$L_2$-stable semigroups, Muckenhoupt weights, and unconditional bases of values of quasi-exponentials

A class of unbounded operators with discrete spectrum in a separable Hilbert space is distinguished, in which the property of being the generator of an $L_2$-stable semigroup is equivalent to the similarity to the Sz.-Nadya–Foiash scalar model. In the proof of this result a connection with the theory of Muckenhoupt weights is established. A criterion for the similarity of a dissipative unicellular operator to the simplest integration operator is also derived. The notion of a quasiexponential, an abstract analogue of an exponential, is introduced. As an application, a description of all unconditional bases in the Hilbert space consisting of values of a quasiexponential is presented.