A. Castejón, E. Corbacho, V. Tarieladze

AMD-Numbers, Compactness, Strict Singularity and the Essential Spectrum of Operators

For an operator T acting from an infinite-dimensional Hilbert space H to a normed space Y we define the upper AMD-number d-(T) and the lower AMD-number d-(T) as the upper and the lower limit of the net (d(T|E)), where E ranges over the family FD(H) of all finite-dimensional subspaces of H. When these numbers are equal, the operator is called AMD-regular.
It is shown that if an operator T is compact, then d-(T) = 0 and, conversely, this property implies the compactness of T provided Y is of cotype 2, but without this requirement may not imply this. Moreover, it is shown that an operator T has the property d-(T) = 0 if and only if it is superstrictly singular. As a consequence, it is established that any superstrictly singular operator from a Hilbert space to a cotype 2 Banach space is compact.
For an operator T, acting between Hilbert spaces, it is shown that d-(T) and d-(T) are respectively the maximal and the minimal elements of the essential spectrum of |T| := (T * T)1/2, and that T is AMD-regular if and only if the essential spectrum of |T| consists of a single point.