By Wolfgang Ruppert

ISBN-10: 3540133879

ISBN-13: 9783540133872

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1 Then (A + B) exists and � � �� �� �� � �� � 1 �� −1 �� �. ��(A + B) �� ≤ ��A−1 �� �� −1 1 − ||A B|| � �� �� The above formula makes sense because ��A−1 B �� < 1. 1, ( �� −1 �� �� −1 �� �� �� ��A B �� ≤ ��A �� ||B|| < ��A−1 �� 1 =1 ||A−1 || ) ( Suppose) (A + B) x = 0. Then 0 = A I + A−1 B x and so since A is one to one, I + A−1 B x = 0. Therefore, ��( �� �� ) �� 0 = �� I + A−1 B x�� ≥ ||x|| − ��A−1 Bx�� �� �� ��) �� ( ≥ ||x|| − ��A−1 B �� ||x|| = 1 − ��A−1 B �� ||x|| > 0 ) ( −1 a contradiction.

Then there exists a sequence xk ∈ X such that � k� �� �� �x � > k ��xk �� , k = 1, 2, · · · . Then deﬁne yk ≡ It follows Letting yik � k� �y � = 1, k xk . |xk | � k� �� �� �y � > k ��yk �� . 4) be the components of y with respect to the given basis, it follows the vector ) ( k y1 , · · · , ynk is a unit vector in Fn . By the Heine Borel theorem, there exists a subsequence, still denoted by k such that ( k ) y1 , · · · , ynk → (y1 , · · · , yn ) . 4 and this that for y= n ∑ yi vi , i=1 �� n �� �� n �� ��∑ �� ��∑ �� �� k �� � � � � � � �� k yi vi �� = �� yi vi �� 0 = lim ��y �� = lim �� �� �� �� k→∞ k→∞ �� i=1 i=1 but not all the yi equal zero.

0 0 Also since U, V are unitary, ( ∗) rank (A∗ ) = rank (V ∗ A∗ U ) = rank (U ∗ AV ) = rank (( σ 0 0 0 )∗ ) = number of singular values. 9 Self Adjoint Operators Approximation In The Frobenius Norm The Frobenius norm is one of many norms for a matrix. It is arguably the most obvious of all norms. Here is its deﬁnition. 1 Let A be a complex m × n matrix. Then ||A||F ≡ (trace (AA∗ )) 1/2 Also this norm comes from the inner product (A, B)F ≡ trace (AB ∗ ) 2 Thus ||A||F is easily seen to equal in Fm×n .

### Compact Semitopological Semigroups: An Intrinsic Theory by Wolfgang Ruppert

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