Abstract. A Helson matrix (also known as a multiplicative Hankel matrix) is an infinite matrix with entries {a(jk)} for j, k ≥ 1. Here the (j, k)'th term depends on the product jk. We study a self-adjoint Helson matrix for a particular sequence a(j) = ( √ j log j(log log j) α )) −1 , j ≥ 3, where α > 0, and prove that it is compact and that its eigenvalues obey the asymptotics λ n ∼ κ(α)/n α as n → ∞, with an explicit constant κ(α). We also establish some intermediate results (of an independent interest) which give a connection between the spectral properties of a Helson matrix and those of its continuous analogue, which we call the integral Helson operator.

We consider a class of maps from integral Hankel operators to Hankel matrices, which we call restriction maps. In the simplest case, such a map is simply a restriction of the integral kernel onto integers. More generally, it is given by an averaging of the kernel with a sufficiently regular weight function. We study the boundedness of restriction maps with respect to the operator norm and the Schatten norms.Of course, for this operation to make sense, the kernel function a has to be continuous. Here is our first result; we denote by S p , 0 < p < ∞, the standard Schatten class of compact operators (see Section 2).

Let G be a locally compact abelian group and let µ be a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in Johansson (Syst Control Lett 57: [105][106][107][108][109][110][111] 2008). We use Laplace transform methods to show that the norm of a convolution operator with symbol µ on such a space is bounded below by the L ∞ norm of the Fourier-Stieltjes transform of µ. We also show that for any Banach lattice of locally integrable functions on G with a shift-invariant norm, the norm of a convolution operator with symbol µ is bounded above by the total variation of µ.Mathematics Subject Classification (2010). Primary 47A30, 47B38; Secondary 43A15.

In this paper we generalise a result of Izuchi and Suárez (K. Izuchi and D. Suárez, Norm-closed invariant subspaces in L ∞ and H ∞ , Glasgow Math. J. 46 (2004), 399-404) on the shift invariant subspaces of L ∞ ‫)ޔ(‬ to the non-commutative setting. Considering these subspaces as C(‫-)ޔ‬modules contained in L ∞ ‫,)ޔ(‬ we show that under some restrictions, a similar description can be given for the B-submodules of A, where A is a C * -algebra and B is a commutative C * -subalgebra of A. We use this to give a description of the ‫ލ‬ n (B)-submodules of ‫ލ‬ n (A).2010 Mathematics Subject Classification. 46L99, 47A15, 47L30.

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