Semigroups of finite convolution operators

Abstract

Semigroups of operators in the commutant A of the Volterra operator J: ƒ(x) → ∝ 0 x f(t) dt on L 2 (0, 1) are constructed and studied by means of a complex Fourier transform technique. A sufficient condition for a fixed operator in the algebra A to be embeddable in a C 0 semigroup of operators in A is given. The condition is shown to be necessary under the further restriction that the semigroup consists of contraction operators (see Theorem 3). In this case, a description of all C 0 semigroups of operators in A containing the fixed operator is obtained (see Theorem 2). Under mild conditions, the domain of analyticity of a semigroup of operators in A is shown to be a sector. Relationships between the lattices of closed invariant subspaces of the operators comprising a semigroup of operators in A and the resolvents of the infinitesimal generator of the semigroup are studied.