A Banach algebra structure for the Wiener algebra \(W(\mathbb{D})\) of the disc. (English) Zbl 1082.47028

It is proved that the Wiener disc algebra \(W(\mathbb{D})\) of all functions holomorphic in the unit disc \(\mathbb{D}\) with the norm \[ \| f\|_{W(\mathbb{D})}=\sum^\infty_{n=0}\,\frac{|f^{(n)}(0)|}{n!} \] with respect to the Duhamel product \[ (f\circledast g)(2)=\frac{d}{dz}\int^2_0 f(z-t)g(t)\,dt \] is a Banach algebra. Characterizations of cyclic vectors for the convolution operators and commutants of the Volterra integration operator are also given.


47B38 Linear operators on function spaces (general)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
32A38 Algebras of holomorphic functions of several complex variables
47A16 Cyclic vectors, hypercyclic and chaotic operators
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