Orthogonality of the range and the kernel of elementary operators.
by
Aleksej Turnsek
Faculty of mechanical engineering, University of ljubljana, Askerceva 6, 1000 ljubljana, Slovenia

Let \phi:B(H) --> B(H) be an elementaty operator defined by \phi(X)=AXB-CXD, where A and C, respectively B and D, are normal commuting operators. We prove that
(i) ||\phi(X)+S|| >= ||S|| for all S in ker\phi and for all X in B(H) if and only if kerA \cap kerC=kerB \cap kerD={0}.

The same characterization also holds (with appropriate assumptions, i.e., S in C_p and \phi(X) in C_p) with respect to the von Neumann-Schatten norms. And in this case we have
(ii) Let p > 1 and kerA \cap kerC=kerB \cap kerD={0}. Then S in ker\phi \cap C_p if and only if ||\phi(X)+S||_p >= ||S||_p for all X in C_p.

For p=2, (M_i)_i=1^k, (N_i)_i=1^k two separately commuting sequences of normal operators and \Delta(X)=\sum_i=1^kM_iXN_i we have
(iii) If \Delta(X) in C₂ and S in ker\Delta \cap C₂, then ||\Delta(X)+S||₂²=||\Delta(X)||₂²+||S||₂².

Finally, some consequences of the above results are also considered.
Note:
These results are from
1. A. Turnsek, Elementary operators and orthogonality, Lin. Algebra Appl., 317 (2000), 207-216.
2. A. Turnsek, Orthogonality in C_p classes, Monatsh. Math., to appear.
3. A. Turnsek, Generalized Anderson's inequality, preprint.

Date received: March 27, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagt-09.