A Class of Operators on L
h
2
. II |
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Authors: | N S Faour |
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Institution: | (1) Department of Mathematics, Lebanese University Faculty of Science I, P. O. Box, 146573 Beirut, Lebanon |
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Abstract: | Let D be the open unit disk in C, and L
h
2
the space of quadratic integrable harmonic functions defined on D. Let
be a function in L (D) with the property that (b) = lim
x
b,x D (x) for all b D. Define the operator C on L
h
2
as follows: C (f) = Q( · f), where Q is the orthogonal projection of L2(D) onto L
h
2
. In this paper it is shown that if C is Fredholm, then is bounded away from zero on a neighborhood of D. Also, if C is compact, then | D 0, and the commutator ideal of (D) is K(D), where (D) denotes the norm closed subalgebra of the algebra of all bounded operators on L
h
2
generated by
, and K(D) is the ideal of compact operators on L
h
2
. Finally, the spectrum of classes of operators defined on L
h
2
is characterized. |
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Keywords: | |
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