For any nonzero invariant subspace M in H2(T2), set . Then Mx is also an invariant subspace of H2(T2) that contains M. If M is of finite codimension in H2(T2) then Mx = H2(T2) and if M = qH2(T2) for some inner function q then Mx = M. In this paper invariant subspaces with Mx = M are studied. If M = q1H2(T2) ∩ q2H2(T2) and q1, q2 are inner functions then Mx = M. However in general this invariant subspace may not be of the form: qH2(T2) for some inner function q. Put (M) = {ø ∈ L ∞: ø M ⊆ H2(T2)}; then (M) is described and (M) = (Mx) is shown. This is the set of all multipliers of M in the title. A necessary and sufficient condition for (M) = H∞(T2) is given. It is noted that the kernel of a Hankel operator is an invariant subspace M with Mx = M. The argument applies to the polydisc case.

In this note, we determine fields K and groups G that are either nilpotent or FC and such that the set of torsion elements of the group ring KG forms a subgroup.

We study certain convexity-type properties of homogeneous functions on topological vector lattices, focusing on a concept of 0+-convexity, and using some probabilistic inequalities.

Necessary and sufficient conditions are found on an ideal a⊲ℤ[x] for the additive group [a]+ of ℤ[x]/a to be finite and cyclic. As a consequence, the abelianizations of certain cyclically-presented groups are computed explicitly.

Arazy has characterized the isometries of p, (0 < p ≦ ∞, p ≠ 2) onto itself as all maps of the form X↦UXV where U and V are either both unitary or both anti-unitary. A simple proof of this result is given.

We establish the Kakutani dichotomy property for two generalized Rademacher–Riesz product measures μ, ν that either μ, ν are equivalent measures or they are mutually singular according as a certain series converges or diverges. We further give sufficient conditions so that in the equivalence case the Radon–Nikodym derivative dμ/dν belongs to Lp(v) for all positive real numbers p, by proving that a certain product martingale converges in Lp(v) for p ≧ 1.

This paper is devoted to the study of rooted properties of phase surfaces defined by complex analytic systems. We first obtain the Rooted Theorem of Analytic Systems. Then we prove the Generalized Strong Rooted Theorem of (m ≧ 2), which implying the Strong Rooted Theorem of a Class of .

Multiresolution is investigated on the basis of shift-invariant spaces. Given a finitely generated shift-invariant subspace S of L2(ℝd), let Sk be the 2k-dilate of S (k∈ℤ). A necessary and sufficient condition is given for the sequence {Sk}k∈ℤ to fom a multiresolution of L2(ℝd). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skew-symmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.

In a previous paper we studied the asymptotic distribution of the multiparameter eigenvalues of uniformly right definite multiparameter Sturm–Liouville eigenvalue problems. In this paper we extend the analysis to deal with multiparameter Sturm–Liouville problems satisfying uniform left definiteness, and non-uniform left and right definiteness.

A combinatorial hypothesis is presented that serves as a necessary and sufficient condition for a union of connected Cockcroft two-complexes to be Cockcroft. This combinatorial hypothesis has a component that can be expressed in terms of the second homology of groups. The hypothesis is applied to the study of the third homology of groups.

Recent work has shown that the solutions of the second-kind integral equation arising from a difference kernel can be expressed in terms of two particular solutions of the equation. This paper establishes analogous results for a wider class of integral operators, which includes the special case of those arising from difference kernels, where the solution of the general case is generated by a finite number of particular cases. The generalisation is achieved by reducing the problem to one of finite rank. Certain non-compact operators, including those arising from Cauchy singular kernels, are amenable to this approach.

We study the influence of the link structure of the prime spectrum of a Noetherian ring on the representation theory of the ring in the case that the ring satisfies the strong second layer condition and has exact integer Gelfand–Kirillov dimension. In particular, we show that Jategaonkar's density condition is satisfied and that the growth of an injective module is controlled by the growth of its first layer.

A problem in descriptive set theory, in which the objects of interest are compact convex sets in linear metric spaces, primarily those having extreme points.