Gap Approximation Numbers and Compactness

Abstract

The gap is considered to be a meaningful metric on the space of all closed operators defined in a Hilbert space . The gap between two closed operators  and  is defined as the gap between the graphs  and  of the operators, which are closed subspaces of the product space . Thus the study of the gap between subspaces has great impact on the gap between operators.

For a bounded operator A, the  approximation number  is defined by  Motivated by this definition, several other approximation numbers were introduced by many Mathematicians for bounded operators. The notion of approximation numbers was further generalized for unbounded operators using the notion of  gap . They are called gap approximation numbers.

In this Paper we analyze connection between compactness and gap approximation numbers.