Eric Nordgren

Invertible Composition Operators on HP

Abstract The weakly closed algebras generated by certain sets of composition operators are shown to be reflexive. A structure theorem for invertible composition operators on H2 is obtained and used to show that such operators are reflexive. The structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts. Another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal. Thus the general invariant subspace problem for Hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators.

On positive linear maps preserving invertibility

THEOREM: A positive linear map 4 between two C*-algebras is a Jordan homomorphism if d preserves invertibility and the range of ) is a C*-algebra. A counterexample is given for the case that the range of Q is not assumed to be a C*algebra; this answers a question raised by B. Russo (Proc. Amer. Math. Sot. 17

Boundary Values of Berezin Symbols

A functional Hilbert space is a collection H of complex-valued functions on some set S such that H is a Hilbert space with respect to the usual vector operations on functions and which has the property that point evaluations are continuous (i.e., for each z ∊ S, the map f → f (z) is a continuous linear functional on H).

A nil algebra of bounded operators on Hilbert space with semisimple norm closure

An algebra of operators having the property of the title is constructed and it is used to give examples related to some recent invariant subspace results.

On invariant operator ranges

A matricial representation is given for the algebra of operators leaving a given dense operator range invariant. It is shown that every operator on an infinite-dimensional Hilbert space has an uncountable family of invariant operator ranges, any two of which intersect only in (0).

An operator not satisfying Lomonosov's hypothesis☆

Abstract An example is presented of a Hilbert space operator such that no non-scalar operator that commutes with it commutes with a non-zero compact operator. This shows that Lomonosovs invariant subspace theorem does not apply to every operator.

Closed densely defined operators commuting with multiplications in a multiplier pair

For a multiplier pair (X,Y ) we study the closed densely defined operators T on X that commute with all of the multiplications by right multipliers in X. We apply our general results to special cases involving Hp, completions of L∞ [0, 1] with respect to certain norms, and the completion of a II1 factor von Neumann algebra with respect to a unitarily invariant norm, where we show that each such T is a “left multiplication”. However, we give an example of a closed densely defined operator on the Bergman space that commutes with all multiplications by H∞-functions but is not a multiplication operator.

On the operator equation AX=XAX

Abstract It is shown that for an operator A on a Hilbert space every solution X of the equation AX=XAX is an idempotent precisely when every restriction of A to an invariant subspace has a dense range.

Invariant Operator Ranges

This report describes a portion of joint work with M. Radjabalipour, H. Radjavi and P. Rosenthal [8]. The set of Hilbert space operators that leave invariant a fixed dense operator range is given a matrical representation. Also it is shown that every operator on an infinite dimensional Hilbert space has an uncountable collection of infinite dimensional invariant operator ranges such that any two of them have only the vector 0 in common.