{"title":"通过de Branges-Rovnyak空间的Cauchy对偶次正规问题","authors":"S. Chavan, S. Ghara, M. R. Reza","doi":"10.4064/sm210419-9-12","DOIUrl":null,"url":null,"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.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2021-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"The Cauchy dual subnormality problem via de Branges–Rovnyak spaces\",\"authors\":\"S. Chavan, S. Ghara, M. R. Reza\",\"doi\":\"10.4064/sm210419-9-12\",\"DOIUrl\":null,\"url\":null,\"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. 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The Cauchy dual subnormality problem via de Branges–Rovnyak spaces
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.
期刊介绍:
The journal publishes original papers in English, French, German and Russian, mainly in functional analysis, abstract methods of mathematical analysis and probability theory.