Operators whose ascent is 0 or
Authors:
P. B. Ramanujan and S. M. Patel
Journal:
Proc. Amer. Math. Soc. 29 (1971), 557560
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 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 finitedimensional operator.
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DOI:
http://dx.doi.org/10.1090/S00029939197102780882
PII:
S 00029939(1971)02780882
Keywords:
Ascent and descent of operators in Hilbert space,
operators with ascent 0 or 1,
direct sum of operators,
compact and finitedimensional operators
Article copyright:
© Copyright 1971
American Mathematical Society
