Two elements Two elements x,y  (xy ¹ yx) of a Banach algebra A such that ex.ey ¹ ex+y.

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Consider A = B(l2) and the unilateral shift operator T on l2, defined by T(x1,x2,...) = (0,x1,x2,...) and its adjoint T*(x1,x2,...) = (x2,x3,...). Assuming xk = (dkn)n Î N,k Î N; < eTeT*x1,x1 > = < eTx1,x1 > = < x1,x1 > = 1, since T*x1 = 0 and Tx1 = x2. Also (T+T*)(x1) = x2, (T+T*)2(x1) = x1+x3, ¼ and so < eT+T*x1,x1 > = < x1,x1 > + < x2,x1 > + < [ 1/ 2!](x1+x3),x1 > +¼ > 1. Hence eT.eT* ¹ eT+T*.


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On 22 Feb 2001, 00:14.