2007
DOI: 10.1007/s00020-007-1502-4
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An Operator-valued Berezin Transform and the Class of n-Hypercontractions

Abstract: We study an operator-valued Berezin transform corresponding to certain standard weighted Bergman spaces of square integrable analytic functions in the unit disc. The study of this operator-valued Berezin transform relates in a natural way to the study of the class of n-hypercontractions on Hilbert space introduced by Agler. To an n-hypercontraction T ∈ L(H) we associate a positive L(H)-valued operator measure dωn,T supported on the closed unit discD in a way that generalizes the above notion of operatorvalued … Show more

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Cited by 27 publications
(20 citation statements)
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“…Full details of this construction can be found in [22,Sections 6 and 7]. We writeÊ =V n (D n,T ) ⊂ E. The mapV n naturally extends to an isometry of A n (D n,T ) into A n (E) with range equal to A n (Ê) by setting…”
Section: Shift Invariant Subspaces In Bergman Spacesmentioning
confidence: 99%
See 3 more Smart Citations
“…Full details of this construction can be found in [22,Sections 6 and 7]. We writeÊ =V n (D n,T ) ⊂ E. The mapV n naturally extends to an isometry of A n (D n,T ) into A n (E) with range equal to A n (Ê) by setting…”
Section: Shift Invariant Subspaces In Bergman Spacesmentioning
confidence: 99%
“…Recall that the operator V n in L(H, A n (D n,T )) is an isometry such that V n T = S * n V n (see [22,Sections 6 and 7]). By formula (1.4) we have that…”
Section: Lemma 23 Let T ∈ L(h) Be An N-hypercontraction In the Classmentioning
confidence: 99%
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“…Nevertheless, the seminal work of Aleman-Richter-Sundberg [2] with later extensions by Shimorin [31] showed that in all cases we recover M as M = span k≥0 S k n (M ⊖ S n M) (a partial analogue of the representation of M in approach 4 above), at least for the cases n = 2, 3. More recently, Olofsson [27,28,29] initiated the study of operator-valued Bergman-inner functions for standard weighted Bergman spaces as an object of independent interest. In particular he suggested a certain time-invariant input/state/output linear system of higher order (past values of the states and of the inputs enter the state-update equations rather than just the current state and input values) as the time-domain explanation for the input-output map corresponding to multiplication by the Bergman inner function.…”
Section: θ(ζ)mentioning
confidence: 99%