ON AN EXAMPLE RELATED TO A CONJECTURE OF RAJENDRA BHATIA AND PETER ? SEMRL

Suppose A and B are two linear operators on Cⁿ with a non-Euclidean norm. In a paper [1] on orthogonality of matrices Bhatia and ? Semrl conjectures that ||A|| ≤ ||A + λB|| for all λ ∈ C iff there exists a unit vector ˜z ∈ C_n such that ||A˜z|| = ||A|| and ||A˜z|| ≤ ||(A + λB)˜z|| for all λ ∈ C. The conjecture was negated by Li [2]. We here give an easy example to negate a slightly modified form of the the conjecture ||A|| < ||A+λB|| for all non-zero scalar λ ∈ C iff there exists a unit vector ˜z ∈ C_n such that ||A˜z|| = ||A|| and ||A˜z|| < ||(A + λB)˜z|| for all non-zero scalar λ ∈ C.