Vol. 52, No. 2, 1974

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Reproducing kernels and operators with a cyclic vector. I

Vashishtha Narayan Singh

Vol. 52 (1974), No. 2, 567–584

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
Received: 27 February 1973
Published: 1 June 1974
Vashishtha Narayan Singh