Operators whose ascent is 0 or
P. B. Ramanujan and S. M. Patel
Proc. Amer. Math. Soc. 29 (1971), 557-560
Proc. Amer. Math. Soc. 34 (1972), 632.
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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.
Brown and Carl
Pearcy, Structure of commutators of operators, Ann. of Math.
(2) 82 (1965), 112–127. MR 0178354
Goldberg, Unbounded linear operators: Theory and applications,
McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0200692
B. Ramanujan, On operators of class (𝑁,𝑘),
Proc. Cambridge Philos. Soc. 68 (1970), 141–142. MR 0261389
E. Taylor, Introduction to functional analysis, John Wiley
& Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
K. Warner, A note on a theorem of Weyl,
Proc. Amer. Math. Soc. 23 (1969), 469–471. MR 0250102
(40 #3343), http://dx.doi.org/10.1090/S0002-9939-1969-0250102-0
- 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)
- S. Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill, New York, 1966. MR 34 #580. MR 0200692 (34:580)
- P. B. Ramanujan, On operators of class (N; k), Proc. Cambridge Philos. Soc. 68 (1970), 141-142. MR 0261389 (41:6004)
- A. E. Taylor, Introduction to functional analysis, Wiley, New York, 1958. MR 20 #5411. MR 0098966 (20:5411)
- 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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Ascent and descent of operators in Hilbert space,
operators with ascent 0 or 1,
direct sum of operators,
compact and finite-dimensional operators
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