Solid extensions of the Cesàro operator on the Hardy space H2(D)H2(D)

We introduce and study the largest Banach space of analytic functions on the unit disc which is solid   for the coefficient-wise order and to which the classical Cesàro operator C:H2→H2C:H2→H2 can be continuously extended, while still maintaining its values in H2H2. Properties of this Banach space H(ces2)H(ces2) as well as a characterization of individual analytic functions which belong to H(ces2)H(ces2) are presented. In addition, both the multiplier space of H(ces2)H(ces2) and the spectrum of C:H(ces2)→H(ces2)C:H(ces2)→H(ces2) are determined.