On certain basis connected with operator and its applications (Q472339)

Let \(B(H,K)\) denote the Banach space of all bounded linear operators between two complex Hilbert spaces \(H\) and \(K\) and let \({\mathcal S}(H)\) be the unit sphere of \(H\). Given an operator \(T\in B(H,K)\), a set \(\{x_k\in {\mathcal S}(H):k\in{\mathcal K}\}\) is called a \(T\)-orthonormal set if \[ x_k\perp x_j\text{ and }Tx_k\perp Tx_j,~k,j\in{\mathcal K},~k\not=j, \] and is called a \(T\)-orthonormal basis if it is a \(T\)-orthonormal set and its linear span is dense in \(H\). The author investigates the existence of such a basis and establishes some permanence properties of it. He then shows that such a basis plays an important role in describing the approximately orthogonality preserving operators, and gives some applications of his results. Furthermore, he answers the question whether a linear operator which approximately preserves orthogonality must be close to an orthogonality preserving one.