This article is cited in 4 papers

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

G. M. Gubreev

South Ukrainian State K. D. Ushynsky Pedagogical University

Abstract: 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.

UDC: 517.986+517.444+517.5

MSC: Primary 47B99, 47B44, 47G10; Secondary 46C10, 42A50

Received: 01.02.1999

DOI: 10.4213/sm442

 English version: Sbornik: Mathematics, 1999, 190:12, 1715–1747

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