Operators whose ascent is 0 or

Authors:
P. B. Ramanujan and S. M. Patel

Journal:
Proc. Amer. Math. Soc. **29** (1971), 557-560

MSC:
Primary 47.10

DOI:
https://doi.org/10.1090/S0002-9939-1971-0278088-2

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 are equal. Let 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 . The main results obtained are the following.

1. The direct sum of a collection of operators is in if and only if each of these operators is in .

2. If and *T* has finite descent, then the range of *T* is closed.

3. If and the range of *T* is not closed, then *T* is a commutator, that is, *T* is expressible in the form for some operators *A* and *B* on *H*.

4. The set of all operators in with descent 0 or 1 is closed in the norm topology of operators.

5. If , and *T* has finite descent and further is compact for some *k*, then *T* is a finite-dimensional operator.

**[1]**A. Brown and C. Pearcy,*Structure of commutators of operators*, Ann. of Math. (2)**82**(1965), 112-127. MR**31**#2612. MR**0178354 (31:2612)****[2]**S. Goldberg,*Unbounded linear operators*:*Theory and applications*, McGraw-Hill, New York, 1966. MR**34**#580. MR**0200692 (34:580)****[3]**P. B. Ramanujan,*On operators of class*(*N*;*k*), Proc. Cambridge Philos. Soc.**68**(1970), 141-142. MR**0261389 (41:6004)****[4]**A. E. Taylor,*Introduction to functional analysis*, Wiley, New York, 1958. MR**20**#5411. MR**0098966 (20:5411)****[5]**K. K. Warner,*A note on a theorem of Weyl*, Proc. Amer. Math. Soc.**23**(1969), 469-471. MR**40**#3343. MR**0250102 (40:3343)**

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DOI:
https://doi.org/10.1090/S0002-9939-1971-0278088-2

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