On Operators Cauchy Dual to 2-Hyperexpansive Operators

Abstract

The operator Cauchy dual to a $2$-hyperexpansive operator $T$, given by $T'\equiv T(T^*T)^{-1}$, turns out to be a hyponormal contraction. This simple observation leads to a structure theorem for the $C^*$-algebra generated by a $2$-hyperexpansion, and a version of the Berger–Shaw theorem for $2$-hyperexpansions.As an application of the hyperexpansivity version of the Berger–Shaw theorem, we show that every analytic $2$-hyperexpansive operator with finite-dimensional cokernel is unitarily equivalent to a compact perturbation of a unilateral shift.

On operators close to isometries

“…In the case where U = U 1 , V = U 2 , and C = C 1,2 (see Example 3.10), the previous corollary recovers Beurling's theorem for Bergman-type operators ( [4], [20]). In the case where U = U 2 , V = U 3 , and C = C 2,3 (Example 3.10), the corollary recovers Beurling's theorem for 2-hyperexpansive operators ( [19], [6]). …”

“…This is one of the reasons one should know more about the operators Cauchy dual to misometries (m ≥ 3) and alternating hyperexpansions ( [20], [6]). …”

“…Hence C 1,2 is a Cauchy map. The fact that C 2,3 is a Cauchy map follows from Theorem 2.9 and Lemma 2.14 of [6].…”

“…It follows from Corollary 2.12 of [6] that T is a hyponormal contraction. Moreover, since the operator Cauchy dual to T is a 2-hyperexpansion ([20, proof of Theorem 3.6]), by Lemma 2.14 and Remark 2.4 of [6], one has σ ap (T ) ⊂ ∂D 1 . This shows that U 1 ⊂ U 3 .…”

Applications of Hilbert Module Approach to Multivariable Operator Theory

“…However, in [DoY92], Douglas and Yan proposed a version of Berger-Shaw theorem in several variables under the assumption that the spectrum of the Hilbert module is contained in an algebraic curve (see also [Zh01]). The reader is also referred to the work of Chavan [Ch07] for a different approach to the Berger-Shaw theorem in the context of 2-hyperexpansive operators. (5) In connection with trace formulae, integral operators, fundamental trace forms and pseudo-differential operators see also Pincus [Pi68], Helton and Howe [HeHo75] and Carey and Pincus [CaPi79], [CaPi77].…”