The Cauchy dual subnormality problem via de Branges–Rovnyak spaces

Abstract

The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a 2-isometry is subnormal. In this paper, we address this problem for cyclic 2-isometries. In view of some recent developments in operator theory on function spaces (see [4, 22]), one may recast CDSP as the problem of subnormality of the Cauchy dual M ′ z of the multiplication operator Mz acting on a de BrangesRovnyak space H(B), where B is a vector-valued rational function. The main result of this paper characterizes the subnormality of M ′ z on H(B) provided B is a vector-valued rational function with simple poles. As an application, we provide affirmative solution to CDSP for the Dirichlettype spaces D(μ) associated with measures μ supported on two antipodal points of the unit circle. 1. Cauchy dual subnormality problem for 2-isometries The Cauchy dual subnormality problem (for short, CDSP) for 2-isometries can be seen as the manifestation of the rich interplay between positive definite and negative definite functions on abelian semigroups. Indeed, CDSP can be considered as the non-commutative variant of the fact from the harmonic analysis on semigroups that the reciprocal of a Bernstein function f : [0,∞) → (0,∞) is completely monotone (see [31, Theorem 3.6]). This fact turns out to be somewhat equivalent to the solution of CDSP for completely hyperexpansive weighted shifts (see [7, Proposition 6] for a generalization). Another early result towards the solution of CDSP asserts that the Cauchy dual of any concave operator is a hyponormal contraction (see [33, Equation (26)]). Later this fact was generalized in [13, Theorem 3.1] by deducing power hyponormality of the Cauchy dual of any concave operator. Around the same time CDSP was settled affirmatively for ∆T -regular 2-isometries in [8, Theorem 3.4] and for 2-isometric operator-valued weighted shifts in [5, Theorems 2.5 and 3.3] (see also [14, Corollary 6.2] for the solution for yet another subclass of 2-isometries). Further, it was shown in [5, Examples 6.6 and 7.10] that there exist 2-isometric weighted shifts on directed trees (that include adjacency operators) whose Cauchy dual is not necessarily subnormal. Recently, a class of cyclic 2-isometric composition operators without subnormal Cauchy dual has been exhibited in [6, Theorem 4.4]. 2000 Mathematics Subject Classification. Primary 47B32, 47B38; Secondary 44A60, 31C25.