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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.

MSC:

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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