**
A. Castejón, E. Corbacho, V. Tarieladze**

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

**abstract:**

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.