On certain basis connected with operator and its applications
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.
approximately orthogonality preserving operators
scientific article; zbMATH DE number 6370940
scientific article