On Besov spaces modelled on Zygmund spaces

Working on the d-torus, we show that Besov spaces Bps(Lp(logL)a) modelled on Zygmund spaces can be described in terms of classical Besov spaces. Several other properties of spaces Bps(Lp(logL)a) are also established. In particular, in the critical case s=d/p, we characterize the embedding of Bpd/p(Lp(logL)a) into the space of continuous functions.

Rota's universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In particular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.