April 2019 Wandering subspace property for homogeneous invariant subspaces
Jörg Eschmeier
Banach J. Math. Anal. 13(2): 486-505 (April 2019). DOI: 10.1215/17358787-2018-0052


For shift-like commuting tuples TB(H)n on graded Hilbert spaces H, we show that each homogeneous invariant subspace M of T has finite index and is generated by its wandering subspace. Under suitable conditions on the grading (Hk)k0 of H, the algebraic direct sum M˜=k0MHk becomes a finitely generated module over the polynomial ring C[z] in n complex variables. We show that the wandering subspace WT(M) of M is contained in M˜ and that each linear basis of WT(M) forms a minimal set of generators for M˜. We describe an algorithm that transforms each set of homogeneous generators of M˜ into a minimal set of generators and can be used to compute minimal sets of generators for homogeneous ideals IC[z]. We prove that each finitely generated γ-graded commuting row contraction TB(H)n admits a finite weak resolution in the sense of Arveson or of Douglas and Misra.


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Jörg Eschmeier. "Wandering subspace property for homogeneous invariant subspaces." Banach J. Math. Anal. 13 (2) 486 - 505, April 2019. https://doi.org/10.1215/17358787-2018-0052


Received: 17 September 2018; Accepted: 27 December 2018; Published: April 2019
First available in Project Euclid: 13 March 2019

zbMATH: 07045469
MathSciNet: MR3927884
Digital Object Identifier: 10.1215/17358787-2018-0052

Primary: 47A13
Secondary: 13P10

Keywords: homogeneous invariant subspaces , minimal sets of generators , wandering subspaces , weak resolutions

Rights: Copyright © 2019 Tusi Mathematical Research Group


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Vol.13 • No. 2 • April 2019
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