On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

• Autorzy: Sameer Chavan
• Języki publikacji: EN

Abstrakty

EN

The Cauchy dual operator T', given by $T(T*T)^{-1}$, provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T' admits the Wold-type decomposition (see Definitions 1.1 and 1.2 below). This connection, which is new even in the bounded case, enables us to establish some interesting properties of unbounded 2-hyperexpansions and their Cauchy dual operators such as the completeness of eigenvectors, the hypercyclicity of scalar multiples, and the wandering subspace property. In particular, certain cyclic 2-hyperexpansions can be modelled as the forward shift ℱ in a reproducing kernel Hilbert space of analytic functions, where the complex polynomials form a core for ℱ. However, unlike unbounded subnormals, $(T*T)^{-1}$ is never compact for unbounded 2-hyperexpansive T. It turns out that the spectral theory of unbounded 2-hyperexpansions is not as satisfactory as that of unbounded subnormal operators.

Kategorie tematyczne

• 47A16: Cyclic vectors, hypercyclic and chaotic operators
• 47A70: (Generalized) eigenfunction expansions; rigged Hilbert spaces
• 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
• 47B33: Composition operators
• 47B20: Subnormal operators, hyponormal operators, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 203
• Numer: 2
• Strony: 129-162

Daty

wydano

2011

Twórcy

autor

Sameer Chavan

• Department of Mathematics & Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India

Identyfikatory

DOI

10.4064/sm203-2-2