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By Abrashkin V.A.

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A nontrivial Q = Q* in Com(T) would yield a nontrivial T-invariant spectral subspace E(A). So irreducibility of T implies Com(T) = {XI}. Conversely, if T has an invariant subspace Yo, then the orthogonal projection P: T-+Y,, gives a nonscalar element of Com(T),QED. Let us remark that for general (nonunitary) T , Schur’s Lemma gives only a necessary condition of irreducibility. It is easy to find examples of reducible T with scalar commutator, for instance, all upper-triangular matrices in Cn (problem 3).

50 that Atl = 0, A t 2 = q. Step 2”. Demonstrate in a similar fashion that “k-transitive*(k+l)-transitive”, completes the argument of Burnside’s Theorem. and The same “bootstrap” applies to transitive algebras A in (oo-D) Hilbert spaces, to show that A is k-transitive for all k! Then fairly standard functional analysis implies that A is weakly closed in %(36) (von Neumann). 9. ). d j = d e g d ; m3. = multiplicity of A’ in T), while the commutator algebja ’ Com(T) 2: e,3 I d j 8 Mat, ’ .. Show, if T N @ ~ @ m ( r ; Tand ) , S2: @ n 8 m ( n ; S ) then , 10.

Spec(Q) is always a c l w d subset of R, and operators with 1-pointed spectrum, spec(&) = {A), are known t o be scalar, Q = XI. Spaces { E ( A ) }are invariant under Q, and have spec(Q I E ( A ) )c A n Z . For operators with discrete spectrum { A k } , space E ( A ) = @ E(Aj),consists of all eigensubspaces with A j E A . An easy way to obtain { E ( A ) ) is to use the canonical model of a selfadjoint operator Q,namely, a multiplication: f(A)-,Af(A), on the space of square-integrable (scalar or vector) functions on C = spec(&), f E Z*(Z;dp).

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2-Divisible groups over Z by Abrashkin V.A.

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