A note on quasi-isometries II

Abstract

An operator A on a complex Hilbert space H is called a quasi-isometry if A 2 A 2 = A A. In the present article, some structural properties of quasi-isometries are established with the help of operator matrix representation. or equivalently,kA 2 xk =kAxk for all x2 H. Obviously the class of quasi- isometries is a simple extension of isometries. The purpose of the present exposition is to explore some properties of quasi-isometries by exploiting the special kind of operator matrix representation associated with such operators. In the course of our investigation, we nd some properties of isometries, which are retained by quasi-isometries. However, there are other ones, which are shown to be false for quasi-isometries. 2. Notations and terminology We use the notations N(A) and R(A) respectively for the null space and the range of an operator A. The symbol F will be used for the closure of a set F. We write (A); 0(A); 00(A);w(A) and W (A) respectively, for the spectrum, the point spectrum, the set of eigenvalues of nite multiplicity, the Weyl spectrum and the numerical range of A. Let r(A) andjW (A)j denote the spectral radius and the numerical radius of A. For an operator A, if w(A) = (A)n 00(A), then we say that the Weyl's theorem holds for A.