Operators whose ascent is $0$ or $1$
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- by P. B. Ramanujan and S. M. Patel PDF
- Proc. Amer. Math. Soc. 29 (1971), 557-560 Request permission
Erratum: Proc. Amer. Math. Soc. 34 (1972), 632.
Abstract:
An operator T on a Hilbert space H is said to be of ascent 0 or 1 if the null spaces of T and ${T^2}$ are equal. Let $\mathcal {A}$ denote the collection of all operators on H which have ascent 0 or 1. The object of this note is to study some properties of operators in $\mathcal {A}$. The main results obtained are the following. 1. The direct sum of a collection of operators is in $\mathcal {A}$ if and only if each of these operators is in $\mathcal {A}$. 2. If $T \in \mathcal {A}$ and T has finite descent, then the range of T is closed. 3. If $T \in \mathcal {A}$ and the range of T is not closed, then T is a commutator, that is, T is expressible in the form $AB - BA$ for some operators A and B on H. 4. The set of all operators in $\mathcal {A}$ with descent 0 or 1 is closed in the norm topology of operators. 5. If $T \in \mathcal {A}$, and T has finite descent and further ${T^k}$ is compact for some k, then T is a finite-dimensional operator.References
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Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 557-560
- MSC: Primary 47.10
- DOI: https://doi.org/10.1090/S0002-9939-1971-0278088-2
- MathSciNet review: 0278088