Journal article
Unbounded operators: Perturbations and commutativity problems
Journal of Functional Analysis, Vol.39(3), pp.281-307
1980
DOI: 10.1016/0022-1236(80)90030-0
Abstract
Let H be a complex Hilbert space, and let G i , i = 1, 2, be closed and orthogonal subspaces of the product space H × H . The subspace G = G 1 ⊕ G 2 is called a (graph) perturbation. We give conditions for invariance of regular operators (R.O.) under graph perturbations: When is the perturbation of a R.O. again a R.O.? If N is a Hilbert space we consider R.O. (i.e., densely defined and closed operators T ) in H = L ( N ) such that G ( T )= G ( S )⊕ V H ( M , where G denotes the graph, S is a decomposable operator in H , V a decomposable partial isometry such that the initial space of V ( t ) is equal to M a.e. t , and finally H ( M ) is the Hardy space of analytic L 2 vector functions with values in M ⊂ N × N . Such operators T commute with the bilateral shift U ; but, unless M = 0, T does not commute with U ∗ . Conversely, this is a canonical model for all R.O. with said commutativity properties. Moreover, the model is unique when T is given, and M = G ( w ) where w is a partial isometry in N . The detailed structure of the model is analyzed in the special case where dim N = dim M = 1. We relate the problem to a condition of Szegő by showing that T is a R.O. iff ∝ 0 2π log ¦ V 2 (t)¦ dt = −∞, where V = (V 1 , V 2 ) is the partial isometry in the special case of dimension one. Szegő's conditions enters in a different way in the analysis of the case M = N × N , as well as in the spectral analysis of T . Our results provide an answer to a commutativity problem posed by Fuglede. If T is an arbitrary densely defined operator, and A ϵ B ( H ) is normal, we prove two theorems stating conditions for the implication A ∪ T ⇒ A ∪ ∗ T . These conditions cannot generally be relaxed.
Details
- Title: Subtitle
- Unbounded operators: Perturbations and commutativity problems
- Creators
- Palle E.T Jorgensen
- Resource Type
- Journal article
- Publication Details
- Journal of Functional Analysis, Vol.39(3), pp.281-307
- DOI
- 10.1016/0022-1236(80)90030-0
- ISSN
- 1096-0783
- eISSN
- 1096-0783
- Language
- English
- Date published
- 1980
- Academic Unit
- Mathematics
- Record Identifier
- 9983985876602771
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