On certain basis connected with operator and its applications

We assume that H,K are the complex Hilbert spaces and a linear operator T∈B(H;K) is given. The object of this paper is to define and consider the certain basis of H connected with the operator T∈B(H;K). This basis will play an important role in describing the approximately orthogonality preserving operators. Next, we will give some application of our results. We give new results and more general ones. Namely, we answer a question, whether a linear operator which approximately preserves orthogonality must be close to an orthogonality preserving one.