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 δ(T ) and the lower AMD-number δ(T ) as the upper and the lower limit of the net (δ(T |E))E∈FD(H) , with respect to the family FD(H) of all finite-dimensional subspaces of H. When these numbers are equal, the operator is called AMDregular. It is shown that if an operator T is compact, then δ(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 δ(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 δ(T ) and δ(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. 2000 Mathematics Subject Classification: Primary: 46C05; Secondary: 47A58, 47A30.