Strong Birkhoff–James Orthogonality of Compact Operators on Hilbert Spaces

In this paper we study strong Birkhoff–James orthogonality and twosided strong Birkhoff–James orthogonality of compact operators. For an infinite-dimensional Hilbert space H, we prove that a compact operator A on H is twosided strongly Birkhoff–James orthogonal to a bounded operator B on H if and only if the rank of the operator \(B^*\) is strictly less than the dimension of the kernel of the operator \(A^*A-\|A\|^2I.\)