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

Sameer Chavan (2011)

Studia Mathematica

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)....

Operator positivity and analytic models of commuting tuples of operators

Monojit Bhattacharjee, Jaydeb Sarkar (2016)

Studia Mathematica

We study analytic models of operators of class C · 0 with natural positivity assumptions. In particular, we prove that for an m-hypercontraction T C · 0 on a Hilbert space , there exist Hilbert spaces and ⁎ and a partially isometric multiplier θ ∈ ℳ (H²(),A²ₘ(⁎)) such that θ = A ² ( ) θ H ² ( ) and T P θ M z | θ , where A²ₘ(⁎) is the ⁎-valued weighted Bergman space and H²() is the -valued Hardy space over the unit disc . We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their applications...

Ordered analytic Hilbert spaces over the unit disk

Shengzhao Hou, Shuyun Wei (2008)

Studia Mathematica

Let f, g be in the analytic function ring Hol(𝔻) over the unit disk 𝔻. We say that f ⪯ g if there exist M > 0 and 0 < r < 1 such that |f(z)| ≤ M|g(z)| whenever r < |z| < 1. Let X be a Hilbert space contained in Hol(𝔻). Then X is called an ordered Hilbert space if f ⪯ g and g ∈ X imply f ∈ X. In this note, we mainly study the connection between an ordered analytic Hilbert space and its reproducing kernel. We also consider when an invariant subspace of the whole space X is similar...

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