About Some Nonsymmetric Operators with Dominant Main Diagonal Normal Operators, Jordan's Blocks
by
Yura Agranovich
Voronezh State Technical University

In article a characterization of the symmetric operators on a finite dimensional Hilbert space wich have a matrix representation with a dominant diagonal with respect to any orthonormal basis are obtained. The main results of this paper contains a generalization for a normal operators case.We shall denote by D_o the set of operators A:H --> H, whose matrices have a dominant main diagonal in any orthonormal basis in n-dimensional Hilbert space H.
Theorem 1. Let A be a nonzero normal operator. Then A in D_o if and only if the inequality
max_{i > j, i, j=[`1, n]}(\surd{\mu_ij} +1/\surd{\mu_ij}) sec (\epsilon_ij/2) <= 2\surd{n/(n-1)},
holds, where \mu_ij = \lambda_i (|A|)/\lambda_j (|A|), \epsilon_ij = arg   \lambda_i (A) - arg \lambda_j (A). 
The proof be based on the following estimate.
Lemma. Let A be a normal operator.
If 0 not in Int{conv[\sigma(A)]} \cup \sigma(A), then the following equality
min _{for allx in H, x =/= 0}\frac|(Ax, x)|||Ax||·||x|| = min _{for all\lambda, \mu in \sigma(A)}\frac2\surd{|\lambda||\mu|}cos(([( /\ ) || (\lambda, \mu)])/2)|\lambda| + |\mu|,
holds.
Corollary 1. Let A in D_o be a normal operator. Then A^*,  | A| in D_o.
So, if A have a matrix representation with a dominant diagonal per lines, then per columnes are valid too, and vice versa.
Corollary 2. Let A in D_o be a nonzero normal operator. Then A^\alpha in D_o for all \alpha in [-1, 1].
Theorem 2. Let A be a Jordan's block 2×2 in some orthonormal basis in H. Then A in D_o(per lines and per columnes) if and only if the inequality sp (|A|² )/|sp  A|² <= (7-4\surd2)/2 is valid.

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Agranovich Yury Ya.: The Theory of Operators with Dominant Main Diagonal.I. Positivity, 1998, 2, No 2 153-164.

Date received: November 19, 1998

Copyright © 1998 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 # cabw-31.