On the Cesaro Operator

On some spaces of holomorphic functions of exponential growth on a half-plane

“…It is interesting to notice that the space H defined in (6), and that in [14] we showed to be equal to M ∆ (A 2 ∆), had already appeared in the literature, in a different context [9,8]. However, H can be described as the closure of polynomials in the L 2 (R, dµ)-norm where dµ(z) = 2 −x |Γ(1 + z)| 2 dA(z), which is not translation invariant in R. In [9] the authors also discussed a Müntz-Szász-type question, concerning the completeness of the powers {(1 − z) λn } in H 2 (D), for λ n > 0 and λ n+1 − λ n > δ; their results however have no (obvious) connection with the Müntz-Szász problem for the Bergman space.…”

“…Therefore, if f is given by (9) it is holomorphic in R and f ∈ H 2 (S ρ n 2 ) for every n, since e ρ n 2 (·) ψ ∈ L 2 (R). It is also clear that f 0 = ψ and arguing as for (10) we obtain (8).…”

H2(μ) spaces and bounded point evaluations

“…To show that extensions of functions in H\β) satisfy (i) and (ii), we refer to the proof of Theorem 8 in [15]. This completes the proof.…”

Composition operators on the Newton space

“…In subsequent papers [5,6,9,11], the methods for constructing the measure were simplified, and additional structural properties of the Newton space were discovered which, in turn, led to additional information about the operator-theoretic properties of the Cesaro operator. In [3] and [10], similar methods are used to uncover information about related operators known as quantum Casaro operators.…”