Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

The idempotent generated subsemigroup of the semigroup of continuous endomorphisms of a separable Hilbert space

R. J. H. Dawlingsa1

a1 Bayero University, PMB 3011, Kano, Nigeria

Let H be a separable Hilbert space and let CL(H) be the semigroup of continuous, linear maps from H to H. Let E+ be the idempotents of CL(H). Let Ker ɑ and Im ɑ be the null-space and range, respectively, of an element ɑ of CL(H) and let St ɑ be the subspace {xH: xɑ = x} of H. It is shown that 〈E+〉 = I∪F∪{i}, where

S0308210500015717_eqnU1

and ι is the identity map. From the proof it is clear that I and F both form subsemigroups of 〈E+〉 and that the depth of I is 3. It is also shown that the depths of F and 〈E+〉 are infinite.

(Received November 06 1981)

(Accepted October 03 1982)