## 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

### Keywords:

Duhamel product; cyclic vector; commutant; Banach algebra
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