Operators of arbitrary large norms that Operators of arbitrary large norms that are bounded by 1 on a given basis of a separable infinite dimensional Hilbert space H.

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Let (xn) be an orthonormal basis for H. For k Î N, define Tk on H by Tkh = < h, x1 + x2 + ¼+ xk > x1. Then
Tkxn = ì
í
î
x1
n £ k
0
n > k
Hence ||Tkxn || £ 1 ( n Î N). On the other hand Tk* h = < h, x1 > (x1 + ¼+xk)  (h Î H); therefore ||Tk || = ||Tk* || ³ ||Tk* x1 || = ||x1 + ¼+ xk || = Ök.

File translated from TEX by TTH, version 2.70.
On 22 Feb 2001, 00:21.