Operators of Almost Hadamard-Type and the Hardy–Littlewood Operator in the Space of Entire Functions of Several Complex Variables

We introduce the class of operators of almost Hadamard type, that is, linear continuous operators that act on a locally convex space containing all polynomials and have the property that the homogeneous polynomials of any given degree form an invariant subspace. The Hadamard type (diagonal) operators, for which each monomial is an eigenvector, are a special case of operators of almost Hadamard type. The operators of almost Hadamard type are studied in the space of all entire functions of several complex variables. The results are used to describe all linear operators continuous in this space and commuting in it with the multidimensional analog of the Hardy–Littlewood operator.