Radial toeplitz operators on the unit ball and slowly oscillating sequences

In the paper we deal with Toeplitz operators acting on the Bergman space A²(Bⁿ) of square integrable analytic functions on the unit ball Bⁿ in Cⁿ. A bounded linear operator acting on the space A²(Bⁿ) is called radial if it commutes with unitary changes of variables. Zhou, Chen, and Dong [9] showed that every radial operator S is diagonal with respect to the standard orthonormal monomial basis (eα)α⊂Nⁿ . Extending their result we prove that the corresponding eigenvalues depend only on the length of multiindex α, i.e. there exists a bounded sequence (λk)∞k=0 of complex numbers such that S eα = λ|α|eα. Toeplitz operator is known to be radial if and only if its generating symbol g is a radial function, i.e., there exists a function a, defined on [0,1], such that g(z) = a(|z|) for almost all z ∈ Bⁿ. In this case Tgeα = γn,a(|α|)eα, where the eigenvalue sequence γn,a(k) ∞k =0 is given by γn,a(k) = 2(k+n) Z 1 0 a(r) r^2k+2n-1 dr = (k+n) ∫ ¹ ₀a(√r) r^k+n-1 dr. Denote by Γn the set {γn,a : a 2 L∞([0,1])}. By a result of Súarez [8], the C-algebra generated by Γ1 coincides with the closure of Γ1 in ̀∞ and is equal to the closure of d1 in ̀∞, where d1 consists of all bounded sequences x = (xk)∞k=0 such that sup k≥0 (k+1) |xk+1 - xk| < +∞. We show that the C*-algebra generated by Γn does not actually depend on n, and coincides with the set of all bounded sequences (xk)∞k =0 that are slowly oscillating in the following sense: |xj - xk| tends to 0 uniformly as j+1/ k+1 → 1 or, in other words, the function x : {0,1,2, . . .} → C is uniformly continuous with respect to the distance ρ( j, k) = |ln( j + 1) - ln(k + 1)|. At the same time we give an example of a complexvalued function a L¹([0,1], r dr) such that its eigenvalue sequence γn,a is bounded but is not slowly oscillating in the indicated sense. This, in particular, implies that a bounded Toeplitz operator having unbounded defining symbol does not necessarily belong to the C-algebra generated by Toeplitz operators with bounded defining symbols.