By P. R. Halmos (auth.), John M. Bachar Jr., Donald W. Hadwin (eds.)

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Extra resources for Hilbert Space Operators: Proceedings, California State University Long Beach Long Beach, California, 20–24 June, 1977

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Thus there are many semi-Fredholm composition operators, but they are either invertible or of index -~. Conditions under which the range of J. Caughran and R. C. Roan. T is one-to-one. hence k z - kw T T(z) = T(w), then CT has dense range only if C~(k z -kw) = 0 C T. (Theorem 1), and Thus we restrict our attention but we will see that this condition alone does not suffice for to be dense. The following is due to Caughran [Ca]. be the range of We can define function It is easy to see that For if T, the range of G is dense have been obtained by is orthogonal to the range of to one-to-one Let CT H2(G) T, so G is a simply connected open subset of by the same method used to define f on G is in H2(G) if and only if function on G.

2] W. Arveson, Operator alsebras and invariant subspaces, Ann. Math. i00 (1974), 433-532. [3] A. S. Besicovitch, A ~eneral form of the coverin~ principle and relative differentiation of additive functions II, Proc. cambridge, Phil. Soc. 42 (1946), 1-10. [4] J. Dixmier, Les algebras d'operateurs dans l'espace Hilbertien, Gauthiervillars, Paris, 1969. [5] P. R. Halmos, Measure Theory, Van Nostrand, New York, 1950. [6] P. R. , 1967. [7] T. L. Kriete, Complete non-self-adjointness of almost self-adjoint operators, Pacific J.

Operators CT, which we will call composition operators, originally appeared in the work of B. O. Koopm~n [Kp] on classical mechanics, as mentioned previously, (see [KN], [N1], [N2], [HN] and [HI]), and they have played a role in ergodie theory (see [H3] ). The initial studies dealt with invertible measure-preserving transformations whereas a number of recent papers have considered more general 40 transformations. B. Examples. properties i. then Before taking up questions of characterization we wish to observe that examples of composition If CT T is an invertible measure-preserving is a unitary operator.

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