It is shown that Lat(, 1/2)(H('(INFIN))(S)), the invariant operator ranges of the commutant of the unilateral shift, is a proper sub-lattice of the lattice of invariant operator ranges of the unilateral shift, S. The notion of a strange operator range for S of order n where n (ELEM) is introduced and it is demonstrated that there exist strange ranges for S of every order. This is done by deriving an operator range condition which is sufficient to insure that a pair of quasi-similar compressions of shifts really be similar. A set of operator ranges which forms a sub-lattice of Lat(, 1/2)(H('(INFIN))(S)) is introduced, which is conjectured to be Lat(, 1/2)(H('(INFIN))(S)). The conjecture is shown to be equivalent to the assertion that the image of S under certain homomorphisms of H('(INFIN))(S) into B(H) is similar to a contraction.;It is proven that any pair of subspaces of a Hilbert space can be the ranges of a pair of commuting operators. A family of one dimen- sional subspaces of a Hilbert space, H, is shown to representable as the set of ranges of a family of commuting operators if and only if for each subspace the linear span of the union of the remaining sub-spaces is not dense in H. The sets of three subspaces of C('3) which can be the ranges of commuting operators are characterized.;It is proven that the ranges of operators from a commutative C*-algebra form a lattice under intersection and vector sum. If P and Q are projections in B(H) with non zero intersection and so that the angle between their ranges is 0, then it is shown that the ranges of the operators in the C*-algebra generated by P and Q does not contain the intersection of the ranges of P and Q. Thus, non-commutative C*-algebras need not have ranges which form a lattice. The question of whether the ranges of operators from different kinds of algebras form lattices is taken up and examples are provided.
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