ABOUT NILPOTENT C0-SEMIGROUPS OF OPERATORS IN THE HILBERT SPACES AND CRITERIA FOR SIMILARITY TO THE INTEGRATION OPERATOR

Abstract. In the paper, we describe a class of operators A that have empty spec-trum and satisfy the nilpotency property of the generated C0-semigroup U(t) = exp{−iAt}, t> 0, and such that the operatorA−1 is similar to the integration oper-ator on the corresponding space L2(0, a). Let U(t), t> 0, be a nilpotent semigroup of C0-class in the separable Hilbert space H. We denote by a the nilpotency index, i.e., U(a) = 0 and U(a1) 6 = 0 for each a1 < a. In what follows, the formula U(t) = exp{−iAt} will mean that the operator −iA is a generator of the C0-semigroup U(t). In such a way the resolvent representation (1) −i(A − zI)−1 = a∫ 0 eiztU(t) dt holds. From this equality, and also from the Wiener-Paley theorem it follows that if −iA generates a C0-semigroup U(t), it will be nilpotent if and only if σ(A) = ∅ and the entire operator-valued function (A − zI)−1 is of the finite exponential type. We’ll get the most simple example of the nilpotent C0-semigroup Ũ(t) = exp{−iAt} if we assume H = L2(0, a) and the inverse to the operator A will be given by formula