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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Operators whose ascent is 0 or $ 1$

Authors: P. B. Ramanujan and S. M. Patel
Journal: Proc. Amer. Math. Soc. 29 (1971), 557-560
MSC: Primary 47.10
Erratum: Proc. Amer. Math. Soc. 34 (1972), 632.
MathSciNet review: 0278088
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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.

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Keywords: Ascent and descent of operators in Hilbert space, operators with ascent 0 or 1, direct sum of operators, compact and finite-dimensional operators
Article copyright: © Copyright 1971 American Mathematical Society

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