The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in the complex plane. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the closed operators that commute with all analytic multiplication operators.

For a multiplier pair ( X , Y ) \left ( X,Y\right ) we study the closed densely defined operators T T on X X that commute with all of the multiplications by right multipliers in X X . We apply our general results to special cases involving H p H^{p} , completions of L ∞ [ 0 , 1 ] L^{\infty }\left [ 0,1\right ] with respect to certain norms, and the completion of a I I 1 II_{1} factor von Neumann algebra with respect to a unitarily invariant norm, where we show that each such T 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 ∞ H^{\infty } -functions but is not a multiplication operator.

The operators on L p ,1 p<∞ that preserve positivity and the constants are shown to have the composition operators as extreme points. In the case of the unit interval with Lebesgue measure they constitute the closed convex hull of these extreme points, but this is not true of all measure spaces.

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II1 factors and Mn(C)) and symmetric gauge norms on L∞[0,1] and Cn. As the first application, we obtain that the class of unitarily invariant norms on a type II1 factor coincides with the class of symmetric gauge norms on L∞[0,1] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286–300] on unitarily invariant norms on Mn(C). As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760–766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative Lp-theory (e.g., non-commutative Hölder's inequality, duality and reflexivity of non-commutative Lp-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N(M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M. We obtain all extreme points of N(M2(C)) and some extreme points of N(Mn(C)) (n⩾3). For a type II1 factor M, we prove that if t (0⩽t⩽1) is a rational number then the Ky Fan tth norm is an extreme point of N(M).

This paper defines a multiplier pair, which is a very general setting in which the notions of multipliers and composition operators can be studied. We prove some results in this general setting, and we present many new interesting examples in which these notions can be interpreted.

Suppose two n-tuples of operators A(k) and B-k on a Hilbert space are given, and F is the mapping from the set of n-tuples of operators on the Hilbert space into the set of all operators on the space defined by Phi(X-1, X-2, ..., X-n) = Sigma(n)(k=1) A(k)X(k)B(k). Conditions are given for Phi to be onto.

A set of matrices S ⊆ Mn(F) is said to be semitransitive if for any two nonzero vectorsx, y ∈ Fn, there exists a matrix A ∈ S such that either Ax = y or Ay = x. In this paper variousproperties of semitransitive linear subspaces of Mn(F) are studied. In particular, it is shown that every semitransitive subspace of matrices has a cyclic vector. Moreover, if |F| ≥ n, it always contains an invertible matrix. It is proved that there are minimal semitransitive matrix spaces without any nontrivial invariant subspace. The structure of minimal semitransitive spaces and triangularizable semitransitive spaces is also studied. Among other results it is shown that every triangularizable semitransitive subspace contains a nonzero nilpotent.

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).

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.