1. CJM Online first
 Chen, Yanni; Hadwin, Don; Liu, Zhe; Nordgren, Eric

A Beurling Theorem for Generalized Hardy Spaces on a Multiply Connected Domain
The object of this paper is to prove a version of the BeurlingHelsonLowdenslager
invariant subspace theorem for operators on certain Banach spaces
of functions on a multiply connected domain in $\mathbb{C}$. 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.
Keywords:Beurling theorem, invariant subspace, generalized Hardy space, gauge norm, multiply connected domain, Forelli projection, innerouter factorization, affiliated operator Categories:47L10, 30H10 

2. CJM 2005 (vol 57 pp. 897)
 Berezhnoĭ, Evgenii I.; Maligranda, Lech

Representation of Banach Ideal Spaces and Factorization of Operators
Representation theorems are proved for Banach ideal spaces with the Fatou property
which are built by the Calder{\'o}nLozanovski\u\i\ construction.
Factorization theorems for operators in spaces more general than the Lebesgue
$L^{p}$ spaces are investigated. It is natural to extend the Gagliardo
theorem on the Schur test and the Rubio de~Francia theorem on factorization of the
Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for
the scales far from $L^{p}$spaces this is impossible. For the concrete integral operators
it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces
are not valid. Representation theorems for the Calder{\'o}nLozanovski\u\i\ construction
are involved in the proofs.
Keywords:Banach ideal spaces, weighted spaces, weight functions,, CalderÃ³nLozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, f Categories:46E30, 46B42, 46B70 

3. CJM 2001 (vol 53 pp. 758)
 Goulden, I. P.; Jackson, D. M.; Latour, F. G.

Inequivalent Transitive Factorizations into Transpositions
The question of counting minimal factorizations of permutations into
transpositions that act transitively on a set has been studied extensively
in the geometrical setting of ramified coverings of the sphere and in the
algebraic setting of symmetric functions.
It is natural, however, from a combinatorial point of view to ask how such
results are affected by counting up to equivalence of factorizations, where
two factorizations are equivalent if they differ only by the interchange of
adjacent factors that commute. We obtain an explicit and elegant result for
the number of such factorizations of permutations with precisely two
factors. The approach used is a combinatorial one that rests on two
constructions.
We believe that this approach, and the combinatorial primitives that have
been developed for the ``cut and join'' analysis, will also assist with the
general case.
Keywords:transitive, transposition, factorization, commutation, cutandjoin Categories:05C38, 15A15, 05A15, 15A18 
