TY - JOUR
T1 - Completely Indecomposable Operators and a Uniqueness Theorem of Cartwright-Levinson Type
AU - Atzmon, A.
AU - Sodin, M.
N1 - Funding Information:
1The second-named author was supported by Grant 96-00030 from the United States Israel Binational Science Foundation.
PY - 1999/12/1
Y1 - 1999/12/1
N2 - A bounded linear operator T on a complex Hilbert space will be called completely indecomposable if its spectrum is not a singleton, and is included in the spectrum of the restrictions of T and T* to any of their nonzero invariant subspaces. Two classes of completely indecomposable operators are constructed. The first consists of essentially selfadjoint operators with spectrum [-2, 2], and the second of bilateral weighted shifts whose spectrum is the unit circle. We do not know whether any of the operators in the first class has a proper invariant subspace and if any of the operators in the second class has a proper hyperinvariant subspace. We also establish a new uniqueness theorem of Cartwright-Levinson type which is the main ingredient in our proofs of complete indecomposability.
AB - A bounded linear operator T on a complex Hilbert space will be called completely indecomposable if its spectrum is not a singleton, and is included in the spectrum of the restrictions of T and T* to any of their nonzero invariant subspaces. Two classes of completely indecomposable operators are constructed. The first consists of essentially selfadjoint operators with spectrum [-2, 2], and the second of bilateral weighted shifts whose spectrum is the unit circle. We do not know whether any of the operators in the first class has a proper invariant subspace and if any of the operators in the second class has a proper hyperinvariant subspace. We also establish a new uniqueness theorem of Cartwright-Levinson type which is the main ingredient in our proofs of complete indecomposability.
UR - http://www.scopus.com/inward/record.url?scp=0001750687&partnerID=8YFLogxK
U2 - 10.1006/jfan.1999.3454
DO - 10.1006/jfan.1999.3454
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AN - SCOPUS:0001750687
SN - 0022-1236
VL - 169
SP - 164
EP - 188
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 1
ER -