Operators whose ascent is 0 or 1

Ramanujan, P. B.; Patel, S. M.

doi:10.1090/S0002-9939-1971-0278088-2

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

Arlen Brown and Carl Pearcy, Structure of commutators of operators, Ann. of Math. (2) 82 (1965), 112–127. MR 178354, DOI 10.2307/1970564
Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0200692
P. B. Ramanujan, On operators of class $(N,\,k)$, Proc. Cambridge Philos. Soc. 68 (1970), 141–142. MR 261389, DOI 10.1017/s0305004100001146
Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
Kenneth K. Warner, A note on a theorem of Weyl, Proc. Amer. Math. Soc. 23 (1969), 469–471. MR 250102, DOI 10.1090/S0002-9939-1969-0250102-0