Reproducing kernels and operators with a cyclic vector. I

Vashishtha Narayan Singh

Vol. 52 (1974), No. 2, 567–584

DOI: 10.2140/pjm.1974.52.567

Abstract

In this paper a study is begun of the complete unitary invariant ((1 − wT)⁻¹e,(1 − zT)⁻¹e), first considered by Livsic in his paper ‘On Spectral Resolution of Linear Nonself Adjoint Operators’ Mat. Sb., 34 (76), 1954, 145–199, of a triple (T,H,e) where T is a bounded linear operator on a Hilbert space H and e is a cyclic vector for T in H, as a reproducing kernel. One of the important points is the construction of a subset of the group algebra of the torus closed under pointwise addition and convolution. This obviously will generate a ring called the K-ring. A study of this ring will be done later.

Several other theorems and constructions are also given.

Mathematical Subject Classification 2000

Primary: 47A65

Secondary: 46E20

Milestones

Received: 27 February 1973

Published: 1 June 1974